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

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426 VII Steady Oseen Fl ow in Exterior Domains

there exists one and only one generalized solutio n to (VII.0.2), (VII.0.3). This

solution satisﬁes the estimates

kvk

2,Ω

+ |v|

1,2

≤ c



R|f|

−1,2

+ (1 + R)kv

∗

1/2,2(∂Ω)



|v(x)| = o(1/

|x|) as |x| → ∞

kpk

2,Ω

/ R

≤ c

{R|f|

−1,2

+ (1 + R)|v|

1,2

}

(VII.2.1 )

for all R > δ(Ω

). In (VII.2.1) p is the pressure ﬁeld associated to v by Lemma

VII.1.1, while c

= c

(R, Ω) (c

→ ∞ as R → ∞).

Proof. We look for a solution of the form

v = w + V

+ σ, (VII.2.2)

where

σ =

4π

∇



|x|



Φ =

∂Ω

∗

· n

(the origin of coordinates has been taken in

Ω

). Further, V

∈ W

1,2

(Ω)

denotes the solenoidal extension of v

∗

−σ|

∂Ω

, of bounded support in Ω, that

was constructed in the proof of Theorem V.1.1. We have

1,2

≤ c kv

∗

1/2,2(∂Ω)

σ = O(1 /|x|

2+|α|

) , |α| = 0, 1, as |x| → ∞.

(VII.2.3 )

Finally, w is requested to be a member of D

1,2

(Ω) and to satisfy the identity

(∇w, ∇ϕ) −R(

∂w

∂x

, ϕ)

= −R[f , ϕ] − (∇V

, ∇ϕ) + R(V

+ σ,

∂ϕ

∂x

(VII.2.4 )

for all ϕ ∈ D(Ω). It is clear that, provided we show the existence of such a

function w, the ﬁeld (VII.2.2) satisﬁes all requirements of generalized solution

to (VII.0 .2), (VII.0.3) given in Deﬁnition VII.1.1. Actually, from (VII.2.3)

and the properties of V

and w, we have v ∈ D

1,2

(Ω); also, v is divergence-

free and assumes the value v

∗

at the boundary. Finally, in view of (VII.2.3)

we have, as |x| → ∞,

|v(x)| ≤

|w(x)| + O(1/|x|

) (VII.2.5)

See ( IV.6.1) for the deﬁnition of the norm involving p.

VII.2 Existence for Three-Dimensional Flow 427

and by Theorem II.7.6 and Lemma II.6.2 we obtain (VII.2.1)

. Thus, to show

the theorem it remains to prove the existence of the ﬁeld w and the vali dity

of estimates (VII.2.1)

1,3

. To this end, l et {ϕ

} be the base of D

1,2

(Ω) deter-

mined in Lemma VII.2.1. We shall construct an “approximate solution” w

to (VII.2.4) in the following way:

`=1

(∇w

, ∇ϕ

) − R(w

∂ϕ

∂x

)

= −R[f , ϕ

] − (∇V

, ∇ϕ

) − R(V

+ σ,

∂ϕ

∂x

) ≡ F

k = 1, 2, . . ., m.

(VII.2.6 )

Using (ii) of Lemma VII.2. 1 we obtain

`=1

(ξ

−Rξ

) = F

, k = 1, 2, . . ., m (VII.2.7)

where

≡ (

∂ϕ

∂x

, ϕ

System (VII.2.7) is linear in the unknowns ξ

, ` = 1, . . ., m, and since A

−A

it is readily seen that the determinant of the coeﬃcients is non-zero. As

a consequence, f or each m ∈ N, system (VII.2.6) admits a uniquely determined

solution. Let us multi ply (VII.2.6)

by ξ

and sum over k from 1 to m. We

obtain

1,2

= −R[f , w

] − (∇V

, ∇w

) −R(V

+ σ,

∂w

∂x

). (VII.2.8)

Using (VII.2.3) and recalling that f ∈ D

−1,2

(Ω), we easily show

−[f, w

] ≤ |f|

−1,2

1,2

−(∇V

, ∇w

) ≤ c

∗

1/2,2(∂Ω)

1,2

−(V

+ σ,

∂w

∂x

) ≤ c

∗

1/2,2(∂Ω)

1,2

and (VII.2.7) furnishes

1,2

≤ c



R|f|

−1,2

+ (1 + R)kv

∗

1/2,2(∂Ω)



. (VII.2.9)

Therefore, the sequence {w

} remains uniformly bounded in D

1,2

(Ω) and,

by Exercise II.6.2, there exist a subsequence, denoted again by {w

}, and a

function w ∈ D

1,2

(Ω) such that in the limit m → ∞

428 VII Steady Oseen Fl ow in Exterior Domains

(∇w

, ∇ϕ) → (∇w, ∇ϕ), for all ϕ ∈ D

1,2

(Ω).

Also, by (VII.2 .9) and Theorem II.2.4 we infer

|w|

1,2

≤ c



R|f|

−1,2

+ (1 + R)kv

∗

1/2,2(∂Ω)



(VII.2.1 0)

with c = c(Ω). For ﬁxed k, we then pass to the limit m → ∞ into (VII.2.6)

deduce wi th no diﬃculty that v satisﬁes (VII.1.1) for all ϕ

. Since, by Lemma

VII.2.1, every ϕ ∈ D(Ω) can be approximated in the W

1,2

-norm by a linear

combination of ϕ

, we establish the validity of (VII.1.1) fo r al l ϕ ∈ D(Ω).

Let us next prove estimates (VII.2.3)

1,3

. We observe that, in view (VII.2.3)

Theorem II.6.1 and the H¨older inequality, we deduce

kvk

2,Ω

≤ |Ω

1/3

kvk

6,Ω

≤ c |Ω

1/3

|v|

1,2

(VII.2.1 1)

where c = c(Ω), and so, since

|v|

1,2

≤ |w|

1,2

+ |V

1,2

+ |σ|

1,2

inequality (VII.2.1)

follows from (VII.2 .10), (VII.2.11), and the properties of

and σ. Let us ﬁnally show (VII.2 .1)

. For ﬁxed R > δ(Ω

), we add to the

pressure p (deﬁned through L emma VII.1.1) the constant

C(R) = −

|Ω

Ω

so that

Ω

(p + C) = 0.

Successively, we take ψ into (VII.1 .2) as a solution to the problem

∇ψ = p + C in Ω

ψ ∈ W

1,2

(Ω

)

kψk

1,2

≤ c

kp + Ck

2,Ω

for som e c

= c

(Ω

). This problem is resolvable in virtue of Theorem II.4.1

and so from (VII.1.2) and the Schwarz inequality we have

kp + Ck

2,Ω

≤ c

(|v|

1,2

+ Rkvk

2,Ω

+ R|f |

−1,2

) (VII.2.12)

which, by (VII.2.11), in turn implies (VII.2.1)

. The solution v j ust con-

structed is unique in view of Theorem VII.1 .2 a nd therefore the proof of

the theorem is accomplished. ut

Remark VI I.2.1 The methods of Theorem VII.2.1 apply with no change

to show existence of solutions in arbitrary space dimension n ≥ 3 , the only

VII.3 The Oseen Fundament al Solution 429

diﬀerence resulting in the asympto tic estimate (VII.2.1)

, which has to be

replaced by

|v(x)| = o(1/|x|

n/2−1

If n = 2, by the same technique we can still establish the exi stence of a

vector ﬁeld v satisfying (i), (ii), (iii), and (v) of Deﬁnition VII.1.1, for q = 2;

however, by this technique we are not able to show the validity of condition

(iv) since, as we know, functions in D

1,2

(Ω) for n = 2 need not tend to a

prescribed value at inﬁnity. Nevertheless, unlike the Stokes approximation,

for the problem at hand we can prove existence of generalized solutions by

means of more complicated tools, as will be shown in Theorem VII.5.1. 

Remark VI I.2.2 The observations made in Remark V.2.1 apply equally to

the present situation. In particula r, if v

∗

= v

+ω×x, for some v

, ω ∈ R

, the

existence of a generalized solution is proved without regularity assumptions

on Ω. 

Exercise VII.2.1 Theorem VII.2.1 can be generalized to the case when ∇·v = g 6≡

0, where g is a suitably prescrib ed function. Speciﬁcally, show that for Ω, f and v

∗

satisfying the same assumptions of Theorem VII.2.1 and for all g ∈ L

(Ω)∩D

−1,2

(Ω)

there exists one and only one generalized solution to the nonhomogeneous Oseen

problem, that is, a ﬁeld v : Ω → R

satisfying (i) (with q = 2), (iii), (iv), and (v)

of Deﬁnition VII.1.1 together with ∇ · v = g in the weak sense. Show, in addition,

that, in such a case, estimate (VII.2.1)

is modiﬁed by adding to its right-hand side

the term

kgk

+ R|g|

−1,2

VII.3 The Oseen Fundamental Solution and the

Associated Volume Potentials

In order to derive further properties of solutions to the Oseen problem in

exterior domains, we shall introduce a suitabl e singular solution to equations

(VII.0.3)

1,2

in the whole space. Though such a solution can be considered, for

the probl em at hand, the analogue of the Stokes fundamental solution (IV.2.3),

(IV.2.4), it diﬀers from this latter in several respects; the main diﬀerence is the

behavior at large distances. Speciﬁcally, the Oseen fundamental solution has

a “no nsym metric” structure, presenting a “wake region” which, as we know,

does not appear i n the Stokes approximation.

Following Oseen (1927, §4), we denote by E and e tensor and vector ﬁelds,

respectively, deﬁned by

430 VII Steady Oseen Fl ow in Exterior Domains

(x, y) =



∆ −

∂

∂y



Φ(x, y)

(x, y) = −

∂

∂y



∆ − 2λ

∂

∂y



Φ(x, y).

(VII.3.1 )

Here i, j = 1, . . . , n, λ = R/2 , while Φ(x, y), with x, y ∈ R

, is any real

function that is smooth for x 6= y. Moreover, the Laplace operator acts on the

y-variables. Observe that if λ = 0 , E and e formally coincide with U and q

introduced in (IV.2.1 ). It i s at once checked that ﬁelds (VII.3.1) satisfy the

following relations for all x 6= y and all i, j = 1, . . ., n



∆ − 2λ

∂

∂y



−

∂

∂y

= δ

∆



∆ − 2λ

∂

∂y



∂

∂y

= 0.

(VII.3.2 )

In order to render (VII.3.1) a singular solution to (VII.0.3)

1,2

, as in the case

of the Stokes system, we choose the function Φ such that

∆



∆ − 2λ

∂

∂x



Φ(x, y) = ∆E(|x −y|), (VII.3.3)

where, we recall, E(x) is the fundamental solutio n (II.9.1) to the Laplace

equation.

A solution to (VII.3.3) is now sought into the form

Φ(x, y) =

2λ

−y

[Φ

(τ, x

− y

, . . . , x

− y

)

−Φ

(τ, x

− y

, . . ., x

− y

)]dτ

(VII.3.4 )

Let L be a (spatial) diﬀerential operator and let h(x, y) be a smooth function of

x, y ∈ R

except at x = y. By the notation

Lh(x, y) = ∆E(|x − y|) (∗)

we mean, as customary, Lh(x, y) = 0 for all x 6= y, while, at x = y, h(x, y)

becomes singular in such a way that for any ψ ∈ C

∞

) it holds that

h(x, y)L

∗

ψ(y)dy = ψ(x),

where L

∗

denotes the formal adjoint of L. Another usually adopted way of writ ing

(∗) is

Lh(x, y) = δ(x − y),

where δ(x) is the symbolic Dirac function, i.e., a distribution deﬁned by the formal

relation

δ(x)ψ(x) = ψ(0)

for all ψ ∈ C

∞

VII.3 The Oseen Fundament al Solution 431

with Φ

and Φ

to be selected appropriately. Replacing formally (VII.3.4) into

(VII.3.3 ) we obtain that Φ

− Φ

must obey

∆



∆ − 2λ

∂

∂y



(Φ

−Φ

) = −2λ∆(

∂E

∂y

). (VII. 3.5)

Choosing

(x, y) = E(|x −y|), (VII.3.6)

for Φ

to be a solution to (VII.3.5) it is suﬃcient to take



∆ − 2λ

∂

∂y



= ∆E. (VII.3.7 )

We notice, in passing, that with the above choice o f Φ, from (VII.3.1)

may take

(x, y) = −

∂

∂y

E(|x − y|), (VII.3.8 )

which shows that the “pressure” e

coincides with the “pressure” q

of the

fundamental Stokes solution, see (IV.2.3)

, (IV.2.4)

. Writing

−λ(x

−y

)

|x − y|

(n−2)/2

f(λ|x − y|), (VII.3.9 )

by a direct computation we deduce



∆ − 2λ

∂

∂y



−λ(x

−y

)

|x − y|

(n+2)/2

(

+ zf

−



n − 2



+ z

)

≡

−λ(x

−y

)

|x − y|

(n+2)/2

L(f),

where z = λ|x − y| and the prime denotes diﬀerentiatio n with respect to

z. Now the equation L(f) = 0 is the well-known Bessel’s modiﬁed equation

which admits two independent soluti ons,

(n−2)/2

(z) and K

(n−2)/2

(z),

called modiﬁed Bessel functions of the ﬁrst and of the second kind, respectively

(MacRobert 1966, §100). However, I

(n−2)/2

(z) is regular for all values of the

argument, while K

(n−2)/2

(z) is singula r at z = 0 in such a way that

(n−2)/2

(z) = log

+ log 2 − γ + σ

(z), if n = 2

(n−2)/2

(z) =

(n−2)/2

Γ (n/2)

n − 2

(n−2)/2

+ σ

(z), if n > 2,

(VII.3.1 0)

432 VII Steady Oseen Fl ow in Exterior Domains

where γ is the Euler constant, Γ is the gamma function, and the remainders

satisfy

(z) = o(1),

= o(z

−k

), k ≥ 1 as z → 0

=o(z

(2−n)/2−k

), k ≥ 0 as z → 0

(Watson 1962, p. 80). Since Φ

must satisfy (VII. 3.7) in the neighborhood

of z = 0, it has to behave there like the fundamental solution E. Thus, with

a view to (II.9.1) and taking into account that ω

= 2π

n/2

/nΓ (n/2), from

(VII.3.9 ) and (VII.3.10) it follows that we must take

= −

2π



2π|x − y|



(n−2)/2

(λ|x − y|)e

λ(y

−x

)

. (VII.3.11)

For n = 3, K

1/2

(z) takes the following simple form (Watson 1962, p. 80):

1/2

(z) =





1/2

−z

, (VII.3.12)

and consequently, from (VII.3.4), (VII. 3.6), and (VII.3.11) we obtain

Φ(x −y) =

8πλ

−y

1 − exp

−λ

+ (x

− y

)

+ (x

−y

)

−τ

+ (x

− y

)

+ (x

−y

)

dτ.

Therefore, ﬁxing the constant up to which Φ is deﬁned by requiring Φ(0) = 0,

it follows that

Φ(x − y) =

8πλ

λ(|x−y|+(x

−y

))

1 − e

−τ

dτ. (VII.3.13)

Correspondingly, the pressure ﬁeld e given in (VII.3.8) takes the form

(x − y) =

4π

− y

|x −y|

. (VII.3.14)

Let us now consider the case n = 2. From (VII.3.4), (VII.3.6) and (VII .3.11)

we ﬁnd

Φ(x − y) =

2λ

−y

[Φ

(τ, x

− y

) −Φ

(τ, x

− y

)] dτ + Φ

−y

)

where

(x − y) =

2π

log |x − y|, Φ

(x − y) = −

2π

(λ|x − y|)e

λ(y

−x

)

and Φ

is a function of x

− y

only, to be ﬁxed appropriately. The function

(z) cannot be expressed in terms of elementary functions; however, we can

provide an asymptotic expansion f or large z:

VII.3 The Oseen Fundament al Solution 433

(z) =





1/2

−z

ν−1

k=0

Γ (k + 1/2)

k!Γ (1/2 − k)

(2z)

−k

+ σ

(z)





1/2

−z



1 −

2!(8z)

+ . . . + σ

(z)



(VII.3.1 5)

where

= O(z

−k−ν

) as z → ∞, k ≥ 0

(Watson 1962, p. 202). In order to choose Φ

, we observe that from (VII.3.6),

(VII.3.7 ) it follows that

∂

∂y

(Φ

−Φ

) = −

∂

∂y



∂

∂y

(Φ

− Φ

) + 2λΦ



and so

∂

∂y

= −

2λ



∂

∂y

(Φ

− Φ

) + 2λΦ



2λ



∂

∂y

(Φ

− Φ

) + 2λΦ



∂

∂y

Since

∂

∂y

log |x −y|



∂

∂y

(λ|x − y|)



= 0,

we conclude

∂

∂y

= −

2λ



∂

∂y

(Φ

−Φ

) + 2λΦ



4π

(λ|x

− y

|) +

∂

∂y

Now, l etting x

−y

→ 0 in this relation, K

(λ|x

−y

|) diverges logarithmi-

cally fast, while the ﬁrst three terms on the right-hand side remain bounded.

This would lead, by (VII.3.1), to an unacceptable singularity for E

, unless

we choose Φ

in such a way that

(t) = −

4π

(λ|t|).

If we do this and impose Φ

(0) = Φ

(0) = 0, we ﬁnd

− y

) = −

4π

−x

− x

− τ)K

(λ|τ|)dτ,

which, in turn, furnishes the following expression for Φ:

434 VII Steady Oseen Fl ow in Exterior Domains

Φ(x − y) =

4πλ

−y

log

+ (x

− y

)



+ (x

−y

)



−λτ



dτ

−

4π

−x

− τ)K

(λ|τ|)dτ.

(VII.3.1 6)

Correspondingly, the pressure ﬁeld e (VII.3.8) becomes

(x − y) =

2π

− y

|x −y|

. (VII.3.17)

The pair E, e deﬁned by (VII.3.1), (VII.3.13), and (VII.3.14) for n = 3 and by

(VII.3.1 ), (VII.3.16), and (VII.3 .17) for n = 2 is called the Oseen fundamental

solution. In arbitrary dimensions n > 3, the Oseen fundamental solutio n is

deﬁned by (VII.3.1), (VII.3.4), (VII.3.8), and (VII.3.11). In view of (VII.3.2)

and (VII.3.3) this solution satisﬁes



∆ − 2λ

∂

∂y



(x − y) =

∂

∂y

(x − y)

∂

∂y

(x − y) = 0

for x 6= y; (VII.3.18)

this is the system adjoint to (VII.0.3). However, since

∂E

∂x

= −

∂E

∂y

∂e

∂x

= −

∂e

∂y

we also have



∆ + 2λ

∂

∂x



(x − y) =

∂

∂x

(x − y)

∂

∂x

(x − y) = 0

for x 6= y (VII.3.19)

with ∆ op erating now on the x-variables.

We wish to investigate the properties of E(x) and e(x) for large |x|. While

those of e are quite obvious, those of E require a little more care. Let us

begin to consider the case where n = 3. Setting r = |x −y|, s = λ(r + x

−y

)

from (VII.3.1)

and (VII.3.13) we derive the following expression for the nine

components o f the tensor E

VII.3 The Oseen Fundament al Solution 435

(x − y) =

4πr



−e

−s

2λr



−y

(1 − e

−s

) + se

−s



(x − y) = −

−s

4πr

8π



−

− y

)



1 − e

−s

+ λ

−s

−1 + e

−s

− y

)



(x − y) = −

−s

4πr

8π



−

− y

)



1 − e

−s

+ λ

−s

−1 + e

−s

− y

)



(x − y) = E

(x − y) =

8πr



− x



−s

−

1 − e

−s

λr



(x − y) = E

(x − y) =

8πr



− x



−s

−

1 − e

−s

λr



(x − y) = E

(x − y) =

8π



−y

)(x

− y

)



−s

−1

+ λr

−s

−1 + e

−s



(VII.3.2 0)

From (VII.3.20) it readily follows that in the limit of vanishing λr the tensor

E reduces to the tensor U (IV.2. 3)

associated to the Stokes fundamental

solution. Speciﬁcally, we have

(x − y) = U

(x − y) + o(1), as λr → 0. (VII.3.21)

In view of (VII.3.21) and (VII.3.1 4), from the calculations leading to (IV.8.15)

we can then show that the Oseen fundamental solution E, e becomes si ngular

at x = y in such a way that, for any vector ﬁeld v continuous at x and all

j = 1, 2, 3, it holds that

lim

ε→0

|x−y|=ε

v · T (w

, e

) · ndσ

= −v

(x), (VII.3.22)

where n is the outer normal to ∂B

(x) and

≡ (E

, E

The estima tes of E(x−y) at inﬁnity are, however, completely diﬀerent from

those of U (x −y). Denote by ϕ the polar angle made by a ray that starts from

x and is directed toward y with the positively directed x

-axis. We present the