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

Подождите немного. Документ загружается.

876 XII Two-Dimensional Flow in Exterior Domains

with

(

3/2 if i = 1

1 if i = 2.

Thus, choosing q ∈ (3, 6), from (XII.8.38), (XII.8.3 9), and (XII.8.40) we ﬁnd

2,2,R/ 2

= O(|x|

−1

). (XII.8.41)

Likewise, for all α ∈ (0, 1),

1,2,R/2

| ≤ c

−1+α

αq,R

,Ω

R/2

(XII.8.4 2)

and so, selecting for arbitrarily sm all ε > 0

α = 1/4 + ε, 3/2α < q < 6,

in view of (XII.8.39), (XII .8.40) a nd (XII.8.42) we obtain

1,2,R/ 2

= O(|x|

−3/4+ε

). (XII.8.43)

The estimates for I

i,3 ,R/2

and I

i,4 ,R/2

are somewhat si mpler. In fact, f rom

(VII.3.4 5) we have

|∇E

(x − y)| ≤ c|x − y|

−3/2

, y ∈ Ω

R/2

, i = 1, 2,

and so, for i = 1, 2,

i,3 ,R/2

| ≤ cR

−3/2

|Ω

R/2

1/q

q,Ω

R/2

i,4,R/2

| ≤ cR

−3/2

|Ω

R/2

1/s

2s,Ω

R/2

By (XII.8.39) we can take q

arbitrarily less than 6 while, by Theorem XII.7.2 ,

s is an arbitrary number greater than 1. Therefore,

i,k,R/2

= O(|x|

−1

), i = 1, 2, k = 3, 4. (XII.8.44)

It remains to estimate the integrals I

R/2

i,k

. By the H¨older inequality and Lemma

XII.8.2, for arbitrarily small η > 0 we have

R/2

i,1

| ≤ ku

2q,Ω

≤ c

∞

R/2

−q+1+η

1/q

≤ c

−1+(2+η)/q

Since, by (XII. 8.35), we can take q

as close to 1 as we please, we deduce

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 877

R/2

i,1

= O(|x|

−1+ε

), i = 1, 2 . (XII.8. 45)

In a similar way, one shows

R/2

i,3

= O(|x|

−1+ε

), i = 1, 2 . (XII.8. 46)

To estimate I

R/2

i,k

, i = 1, 2, k = 2, 4 we argue as follows. For α ∈ (0, 1), by

Lemma XII.8.2 and the H¨older inequal ity we have for arbitrarily small η > 0

R/2

i,2

| ≤ c

−1+α+η

2 ku

αq

. (XII.8.4 7)

In view of (XII.8.39) and (XII.8.40 ), for i = 2 we may take α arbitrarily close

to zero to deduce, for any ε > 0, that

R/2

2,2

= O(|x|

−1+ε

). (XII.8.48)

On the other hand, if i = 1, we choose α = 2/5 + δ, arbitrarily small δ > 0,

and q ∈ (6/5α, 3), so that, again by (XII.8.39) and (XII.8.40), the inequality

(XII.8.4 7) delivers

R/2

1,2

= O(|x|

−3/5+ε

). (XII.8.49)

In a completely analogous way we obtain

R/2

2,4

= O(|x|

−1+ε

) (XII.8. 50)

and

R/2

1,4

= O(|x|

−3/5+ε

). (XII.8.51)

We shall now show that estimates (XII.8.49), (XII. 8.51) can, in fact, be im-

proved. Actuall y, from (XII.7.27 )–(XII.7.2 9), (XII.8.32), (XII.8.36), (XII.8 .41),

(XII.8.43)–(XII.8.46),and (XII.8.48)–(XII.8.51) we deduce the followingasymp-

totic uniform bounds for u:

(x) = O(|x|

−1/2

), u

(x) = O(|x|

−1+ε

). (XII.8.52)

Let us now spl it the integral I

R/2

1,2

as the sum of two integrals: one over Ω

R/2,2R

and the other over Ω

. Let us denote them by I

and I

, respectively. In

view of (XII.8.52), for a rbitrarily small ε > 0, we have

| ≤ cR

−3/2+ε

k∇E

1,B

(x)

and so the estimate of Exercise VII.3. 4 implies

| ≤ c

−3/2+ε

−1/2

dr + 1

which, in turn, gives

= O(|x|

−1+ε

). (XII.8.53)

878 XII Two-Dimensional Flow in Exterior Domains

Consider, next, I

. For |y| ≥ 2R = 2|x|, by the triangle inequality it follows

that

2|y| ≥ |x − y|

and so, with the aid of (XII.8.5 2) and Exercise VII.3.4, we ﬁnd for arbitrarily

small ε > 0 that

| ≤ c

Ω

|x −y|

−3/2+ε

|∇E

(x − y)|dy

≤ c

(x)

|x −y|

−3/2+ε

|∇E

(x − y)|dy

≤ c

∞

−2+ε

dρ.

Therefore

= O(|x|

−1+ε

). (XII.8.54)

From (XII. 8.53), (XII. 8.54) we infer

R/2

1,2

= O(|x|

−1+ε

). (XII.8.55)

In an entirely analogous way we show

R/2

1,4

= O(|x|

−1+ε

). (XII.8.56)

To complete the proof of the theorem we need to improve the estimate on the

term I

1,2,R/2

given in (XII.8.43). From what we have proved so far we know

that u admits the following asymptotic estimates

(x) = m · E

(x) + n

[u(x)] + O(|x|

−1

)

(x) = O(|x|

−1+ε

(XII.8.5 7)

where

= I

1,2,R/2

+ O(|x|

−1+ε

). (XII.8.58)

Recalling the expression of I

1,2,R/2

and using (XII.8.43), (XII.8.57), and

(XII.8.5 8) we deduce that

1,2,R/ 2

| ≤ c

Ω

R/2

g(y) (|n

(y)| + |E

(y)| + g(y)) |D

(x − y)|dy,

where g satisﬁes the assumptio ns of Lemma XII. 8.1. Using this lemma along

with the H¨older inequality, from the preceding inequality we deduce

1,2,R/ 2

| ≤ c

Ω

R/2

g(y)|n

(y)||D

(x − y)|dy + O(|x|

−1+ε

). (XII. 8.59)

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 879

In view of (XII.8.58), again employing Lemma XI I.8.1 we have

1,2,R/2

| ≤ c

Ω

R/2

g(y)|I

1,2,R/2

(y)||D

(x − y)|dy + O(|x|

−1+ε

Now assume that for some β ∈ [3/4, 1),

1,2,R/ 2

(x)| ≤ c

|x|

−β +ε

. (XII.8 .60)

Recalling that g satisﬁes the assumptions of Lemma XII. 8.1 we ﬁnd

1,2,R/ 2

| ≤ c

−1+α

αq

−1−β+2/q

+ε

+ 1] + O(|x|

−1+ε

). (XII.8.61)

We want the quantity in square brackets to be bo unded in R and so we require

1 + β −2/q

> 0.

Recalling that q is a number strictly greater than 3/2α which can be taken

arbitrarily close to 3/2 α (cf. (XII.8.40), (XII.8.42)), we thus ﬁnd

1 − β = 4α/3 −η, (XII.8.62)

for arbitrarily small positive η. With this restriction for α, from (XII.8.61) we

obtain a new value for β, that is,

= 1 −α,

for whi ch (XII.8.60) holds. We may iterate this procedure along sequences

{β

}, {α

} which, by virtue of this last condition, (XII.8.43), and (XII.8.62),

satisfy

1 − β

= 4α

/3 − η, β

k+1

= 1 − α

, β

= 3/4.

Solving for β

we see

k+1

= 1/4 + 3β

/4 − 3η/4.

Since { β

} is increasing, by the arbitrariness of η, from this relati on we ﬁnd

→ 1 as k → ∞, and from (XII.8.60) we conclude

1,2,R/ 2

= O(|x|

−1+ε

which completes the proof of the theorem. ut

Remark XI I.8.1 As in the three-dimensional situation, from Theorem XII.8.1

we obtain that the component v

parallel to the velocity at inﬁnity v

∞

= (1, 0)

exhibits a paraboloidal wake behavior in the direction of v

∞

. Actually, for all

suﬃciently large |x| we obtain the uniform estimate

Throughout the rest of the proof, the symbol ε in the various formulas need not be

the same. However, it will denote, as usual, an arbitrarily small positive number.

880 XII Two-Dimensional Flow in Exterior Domains

(x) + 1 = O(|x|

−1/2

Moreover, let ϕ be the angle made by a ray starting from the origin (in Ω

say) with the negatively directed x

-axis, and deﬁne the parabo lic-l ike region:



(x, ϕ) ∈ R

×(−π/2, π/2) : (1 + cos ϕ) ≤ | x|

−1+2σ

, σ ∈ (0, 1/2]



(XII.8.6 3)

Then, in view of (VII.3.40), (VII.3.41), we have the faster decay

(x) + 1 = O(|x|

−1/2 −α

) , x 6∈ P

, (XII.8.64 )

with α = min(2σ, 1/2 − ε), arbitrary small ε > 0. Estimate (XII.8.64) can

be slightly improved, due to the fact tha t estimate (XII.8.31 ) is not the best

possible. A detailed study in this direction has been performed by Smith

(1965), who shows sharp bounds for the nonlinear term and, therefore, for the

remainder V deﬁned in (XII.8.30). In particular, (XII.8.30) can be improved

to the following:

V(x) = O



|x|

−1

log

|x|



Sharper estimates for V

can also be obtained in the region (XII.8.63),

cf. Smith (1965, Theorem 5). Speciﬁcally, in this regio n and for large |x|, one

can prove

(x) = O



|x|

−1−σ

log |x|(1 + |x|

−σ

log |x|)



On the other hand, unlike the three-dimensional case, the component v

of v

orthogonal to v

∞

“essentially” exhibits no wake region. In fact, from Theorem

XII.8.1 we have that for all suﬃciently large |x|, inside and outside the wake

region,

(x) = O(|x|

−1+ε

It should be observed that this uniform estima te can be slightly improved to

the following:

(x) = O



|x|

−1

log |x|



as shown by Smith (1965, formula (28b)). Sharper estimates for the remainder

can be proved in the region (XII.8.63), cf. Smith (1965, Theorem 5), where

we have for |x| large enough,

(x) = O



|x|

−1−σ

(1 + |x|

−σ

log |x|)



An asymptotic expansion for v up to |x|

−3/2

has been given by Babenko (1970,

Theorem 6.1). 

Remark XI I.8.2 The uniform estimate

v(x) + v

∞

= O(|x|

−1/2

)

is sharp in the sense that the condition

See Remark VII .3.1.

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 881

v(x) + v

∞

= o(|x|

−1/2

) (XII.8 .65)

holds if and only if we impose some restriction on the ﬂow. Speciﬁcally, we

have that (XII. 8.65) is valid if and only if the ﬁrst component of the vector

m deﬁned in (XII.8.29) is vanishing. The proof foll ows the same lines as that

given in Corollary X.8.1 for the three-dimensional case and i s therefore left to

the reader as an exercise.

This result diﬀers from the correspondi ng one in

three dimensions where v(x) + v

∞

= o(|x|

−1

) if and only if all components

of m are zero. The fact that in two dimensions only m

needs to be zero is a

consequence of the property that, in such a case, the component E

of the

Oseen fundamental tensor exhibits no wake. 

Our next (and ﬁnal) task in this section is to study the behavior at inﬁnity

of the ﬁrst derivatives of v and of the pressure ﬁeld p. This study is performed

along the same lines as the proof of Theorem XII.8.1. From the representation

formulas (XII.7.27)

–(XII.7. 29) we obtain for i, k = 1, 2 and all suﬃciently

large |x|

(x) = M · D

(x) + RN

i,k

[u(x)] + D

(2)

(x), (XII.8.66)

with E

deﬁned in (XII.5.17) and

i,k

[u(x)] =

Ω

u(y) · ∇u(y) · D

(x − y)dy. (XII. 8.67)

As in Theorem XII.8 .1, the asymptotic behavior of ∇v is determined once

we establish that for N

i,k

. It is expected that the behavior of D

v will be

diﬀerent for diﬀerent values of k, as a consequence of the unequal behavior at

large distances of D

E, cf. (VII. 3.45) and (VII.3.46). However, the method of

proof is essentially the same for both k = 1, 2 and, therefore, we shall restrict

our attention to k = 1, limiting ourselves to state the result for k = 2 in

Theorem XII.8.2. Setting |x| = R, we split N

i,1

as the sum of two integrals

i,R/2

and I

R/2

on the do mai ns Ω

R/2

and Ω

R/2

, respectively. By (VII.3.45)

we have, for i = 1, 2,

(x − y)| ≤ c|x − y|

−3/2

, as |x − y| → ∞,

with E

deﬁned in (XII.5.17 ). On the other hand, by Theorem XII.7.2 we also

have

u · ∇u ∈ L

(Ω), for all q > 1, (XII.8.68)

so that we deduce

i,R/2

| ≤ c|x|

−3/2

|Ω

R/2

−1/q

ku · ∇uk

≤ c

|x|

−3/2

|Ω

R/2

−1/q

Taking q

arbitrarily large, we i nfer

See also Smith (1965, Theorem 11).

882 XII Two-Dimensional Flow in Exterior Domains

i,R/2

= O(|x|

−3/2+ε

), i = 1, 2, (XII.8.69)

where ε > 0 is arbitrarily small. We now pass to estimate I

R/2

. From Theorem

XII.8.1 and with the help of the local representation (IX.8.33) we show

v(x) = O(|x|

−1/2

), k = 1, 2. (XII.8.70)

We have

R/2

Ω

R/2

(y)D

u(y) ·D

(x − y)

(y)D

u(y) · D

(x − y)]dy

≡ I

+ I

From Theorem XII.8.1 and (XII.8.69) it follows for q < 2 that

| ≤ c

Ω

R/2

|y|

−1

(x − y)|dy

≤ c

−1+2/q

q,R

| ≤ c

Ω

R/2

|y|

−3/2+ε

(x −y)|dy

≤ c

−3/2+(2+ε)/q

q,R

In view of (XII.8.35), we may take in these relations q

arbitrarily l arge to

ﬁnd

= O(|x|

−1+ε

)

= O(|x|

−3/2+ε

(XII.8.7 1)

From (XII. 8.66), (XII. 8.67), (XII. 8.69), and (XII.8.71) we obtain

v(x) = O(|x|

−1+ε

), k = 1, 2,

which improves on (XII.8.69) with k = 1. We may then use this la tter estimate

in bounding the integral I

. If we do this and proceed as before, we ﬁnd

| ≤ c

Ω

R/2

|y|

−3/2+ε

(x − y)|dy

which, together with the H¨o lder inequality and (XII.8.35), in turn, implies

= O(|x|

−3/2+ε

We may then conclude

XII.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 883

i,1

[u(x)] = O( |x|

−3/2+ε

), i = 1, 2, (XII.8.72)

for a rbitrary ε > 0. In a completely analogous way we show

i,2

[u(x)] = O( |x|

−1+ε

), i = 1, 2. (XII.8.73)

We thus have proved the fol lowing.

Theorem XII.8.2 Let v be as in Theorem XII.8.1. Then as |x| → ∞,

v(x) = M ·D

E(x) + T

(x), k = 1, 2,

holds, where (i = 1, 2)

= −

∂Ω

(u, p) + Rδ

+ R

Ω

E is the Oseen fundamental tensor, and

(x) = O(|x|

−α

+ε

with α

= 3/2 a nd α

= 1.

Remark XI I.8.3 Sl ightly improved uniform estimates can b e given for the

remainder T

(x); see Smith (1965). Speciﬁcally, for large |x|, one has the

uniform estimate

(x) = O(|x|

−1

log

|x|)

(x), T

2,2

= O(|x|

−3/2

log

|x|)

(x) = O(|x|

−3/2

Outside the wake region (XII.8.63), o ne can prove the following sharper esti-

mates

(x) = O(|x|

−1−2σ

log

|x|)

(x), T

2,2

= O(|x|

−3/2 −σ

log

|x|)

(x) = O(|x|

−3/2 −σ

For detail s, we refer the interested reader to Theorem 6 of Smith (1965 ). 

Our next objective is to derive an asymptotic formula for the pressure

ﬁeld. To this end, we start with the representation (XII.7.27)

, which can be

written as

p(x) = p

−M

∗

(x) − R

k=1

(x) + h(x) (XII.8.74)

where

884 XII Two-Dimensional Flow in Exterior Domains

(x) =

Ω

R/2

(x − y)u

(y)D

(y)dy

(x) =

Ω

R/2,2R

(x − y)u

(y)D

(y)dy

(x) =

Ω

(x − y)u

(y)D

(y)dy

(XII.8.7 5)

and R = |x|. Furthermore, the q uantities M

∗

, h are deﬁned in (XII.7 .23) a nd

(XII.7.2 4). Using the uniform estimate

|e(x −y)| ≤ c|x − y|

−1

together with (XII.8.68), Theorem XII.8.1 , Theorem XII.8.2, and Lemma

II.9.1, we deduce, with ε meaning a n arbitrary positive number, that

(x)| ≤ c

−1

|Ω

R/2

1/q

ku · ∇uk

≤ c

|x|

−1+ε

(x)| ≤

Ω

R/2,2R

(|u

(y)D

(y)| + |u

(y)D

u(y)|) |e(x − y)|dy

≤ c

|x|

2−ε

Ω

R/2,2R

|e(x −y)|dy ≤ c

|x|

−1+ε

(x)| ≤ c

Ω

|y|

2−ε

|x −y|

−1

dy ≤ c

|x|

−1+ε

(XII.8.7 6)

Collecting (XII.7.24)

, and (XII.8.74)–(XII.8 .76) furnishes the following.

Theorem XII.8.3 Let v be as in Theorem XII.8.1. Then there is a p

∈ R

such that as |x| → ∞,

p(x) = p

− M

∗

(x) + P(x),

where (i = 1, 2)

∗

= −

∂Ω

(u, p) + R[δ

−δ

]}n

+ R

Ω

e(x) is the pressure associated to the Oseen fundamental tensor E, and

P(x) = O(| x|

−1+ε

)

for a rbitrary ε > 0.

Remark XI I.8.4 A minor improvement on the behavior of the term P(x)

can be obtained starting with a representation formula slightly diﬀerent from

(XII.8.7 4); cf. Smith (19 65, Theorem 7). 

XII.9 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem 885

Finally, we wish to mention some properties regarding the asymptotic

structure of the vorticity ﬁeld:

ω =

∂v

∂x

−

∂v

∂x

(XII.8.7 7)

associated to a solution v to (XII.0.1), (XII.0.2) in the case v

∞

6= 0. This

problem has been studied in full detail by Babenko (1970 ) and Clark (1 971).

The main result states, essentially, that if f is of bounded support and if v

satisﬁes an estimate of the form

v(x) + v

∞

= O(|x|

−1/4−ε

) (XII.8.78)

for some ε > 0, then ω decays exponentially fast in the region outside the

paraboloidal wake region. Now, as we obtain from Lemma XII.8.2, every gen-

eralized solution that tends pointwise and unif ormly to v

∞

(6= 0) at inﬁnity

obeys (XII.8.78), and so the vorticity ﬁeld of every such generalized solution

corresponding satisﬁes the above mentioned property. In particular, this holds

for solutions determined in Theorem XII. 5.1. Thus from Lemma XII.8.2 and a

theorem of Babenko (1970, Theorem 8.1) (cf. also Cla rk 1971, Theorem 3.5

one can show the following result, whose proof will be omitted.

Theorem XII.8.4 Let the assumptions of Theorem XII.8.1 be satisﬁed.

Then the vorticity ﬁeld (XII.8.77 ) obeys the following representation for all

suﬃciently large |x|

ω(x) = ∇Ψ(x) × m + B(x),

where

Ψ(x) ≡ e

(R|x|/2) ,

is the modiﬁed Bessel function of the second type of order zero (cf. (VII.3.10),

and (VII.3.15)), m is deﬁned in (XII.8.29), and

B(x) = O(e

−s(x)

|x|

−3/2

log |x|), s(x) := (|x| + x

XII.9 Limit of Vanishing Reynolds Number: Transition

to the Stokes Problem

In this section we shall investigate the behavior of solutions constructed in

Theorem XII.5.1, in the limit of vanishing Reynolds number. We shall prove, in

particular, that they converge to a uniquely determined solution of a suitable

As usual, if a = (a

, a

) and b = (b

, b

), by the notation a × b we mean the

quantity a

−a