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

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436 VII Steady Oseen Fl ow in Exterior Domains

estimates for E(x −y) as a function of y for ﬁxed x. Considered as a function

of x, all estimates remain true if ϕ is replaced by π − ϕ, see Remark VII.3.1.

Taking x as the origin of coordinates (this produces no loss of generality since

E is a function of x − y only) and noticing that

−s

≤ (1 −e

−s

)/s, s > 0, |(y

− x

)/s| ≤ 2/λ, j = 2, 3 ,

from (VII.3.20) we o bta in

|E(y)| ≤

|y|

1 − e

−s

, (VII.3.23)

where c

= c

(λ) and

s = λ(|y| −y

) = λ|y|(1 −cos ϕ).

The bound (VII.3.23) f urnishes, in particular, the uniform estimate

|E(y)| ≤

|y|

, (VII.3.24 )

which coincides with tha t given for U in (IV.2.6). However, improved bounds

can b e derived from (VII.3 .23) as a function of ϕ. Speciﬁcally, if

(1 − cos ϕ) ≥ |y|

−1+2σ

for some σ ∈ [0, 1/2], (VII.3.25)

then (VII.3.23) implies

|E(y)| ≤

|y|

1+2σ

, (VII.3.26 )

with c

= c

(λ). From (VII.3.23)–(VII.3.26) it follows tha t if y belongs to the

region deﬁned by

|y|(1 − cos ϕ) ≤ 1 (VII.3.27)

then (VII.3.24) holds. This region represents a parabo loid with the axis in the

direction of the negative x

-axis and can be interpreted as a “wake.”

Remark VI I.3.1 The tensor E(x − y) considered as a function of y thus

exhibits a “wake” region in the direction opposite to what is expected for a

body moving in a liquid with the velocity v

directed in the positive x

-axis

(as we have assumed at the beginning of the chapter). This is due to the fact

that, as a function of y, E satisﬁes the adjoint system (VII.3.18). However,

if we consider E as a function of x, then ϕ should be changed in π − ϕ and

the parabolo idal wake region becomes appropriately located with its axis in

the direction of the negative x

-axis. This remark is important in the context

of the a symptotic structure of solutions to the Oseen system; see Theorem

VII.6.2 and Remark VII.6.1. 

VII.3 The Oseen Fundament al Solution 437

Starting from (VII.3.23) we may also derive the summability properties

of E(y) in the exterior A of a ball of unit (say) radius. In particular, by a

straightforward calculation we show

E(y) ∈ L

(A) for all q > 2. (VII.3.28)

This estimate is sharp; a ctual ly, from (VII.3 .20)

we have with a = 1/λr

(y)| =



8πr



−s

[1 + (1 − a) cos ϕ] + a cos ϕ|

Thus, for all r ≥ λ and all ϕ ∈ [0, π/2], we ﬁnd

(y)| ≥

8πr

−λr(1−cos ϕ)

From this relation it follows tha t

q,A

≥ c

∞

−q+2

−λqr

π/2

λqr cos ϕ

sin ϕ dϕ

= c

∞

−q+2

−λqr



λqrx



= c

∞

−q+1

(1 −e

−λqr

) dr .

(VII.3.2 9)

Therefore, setting

R = max



λ, −

λq





from (VII.3.29) we o bta in

q,A

≥ c

∞

−q+1

dr,

showing that E

does not belong to L

(A) fo r all q ∈ [1, 2 ]. A similar con-

clusion can be reached for the other components of E, by means of the same

technique, which allows us to conclude

E(y) 6∈ L

(A), for all q ∈ [1, 2]. (VII.3.30)

The summability properties (VII.3.28), (VII. 3.30) should be contrasted with

the analogous properties of the Stokes fundamental tensor U, for which (in

dimension 3) we have that U belongs to L

(A) for all and only all q > 3.

Estimates similar to (VI I.3.23) can be derived for ﬁrst derivatives. Specif-

ically, from (VII.3.20) the following inequalities are directly obtained

438 VII Steady Oseen Fl ow in Exterior Domains



∂E(y)

∂y



≤

|y|

3/2



1 − e

−s

− se

−s

3/2

|y|

1/2

1 − e

−s



, i = 2, 3



∂E(y)

∂y



≤

|y|

1 − e

−s

(VII.3.3 1)

with c = c(λ). These formulas imply, in particular, the uniform bound for |y|

greater than any ﬁxed R

> 0:

|∇E(y)| ≤

|y|

3/2

(VII.3.3 2)

with c

= c

(λ, R

). However, better estimates can be derived outside the

“wake region” (VII. 3.27), in a way completely analog ous to that used previ-

ously, and we leave them to the reader a s an exercise. Concerning the summa-

bility of ∇E in a neighborhood of inﬁnity, we notice that by (VII.3.31) we

have

∂E(y)

∂y

∈ L

(A), for all q > 4/3, i = 2, 3

∂E

∂y

∈ L

(A), for all q > 1,

(VII.3.3 3)

where A, as before, denotes the exterior of a uni t ball. Condition (VII.3.33)

is sharp in the sense that

∂E(y)

∂y

6∈ L

(A), for all q ∈ [1, 4/3], i = 2, 3 (VII. 3.34)

as a result o f a calculation similar to that used to show (VII.3.30). It is inter-

esting to observe that, even though the unif orm bo und (VII.3.32) is weaker

than the analogous one for the Stokes fundamental tensor U (see (IV. 2.6)),

the summability properties (VII.3.33) are stronger than those for U , where

(in dimension 3) ∇U ∈ L

(A) for a ll and only all q > 3/2.

Further estima tes can be obtained for derivatives of order higher than one.

Here, we shall limit ourselves to presenting some bounds of particula r interest,

leaving their proofs to the reader.



∂

E(y)

∂y



≤

|y|



1 − e

−s

−se

−s

|y|

1 − e

−s



, y ∈ R

− {0}

E(y)| ≤ c|y|

−1−|α|/2

, |α| ≥ 2 , suﬃciently large |y|.

(VII.3.3 5)

Exercise VII.3.1 Show that (in three dimensions) the tensor E satisﬁes the fol-

lowing estimates for all R > 0:

VII.3 The Oseen Fundament al Solution 439

∂B

|E(y)|

≤ cR

−1

∂B

|∇E( y)| ≤ cR

−1/2

∂B

|∇∇E(y)| ≤ cR

−1

Hint: Use estimates (VII.3.23), (VII.3.33), and (VII.3.35); see also Solonnikov (1996).

We next derive the form and corresponding estimates for E(x −y) in two

space dimensions. From (VII.3.1) and (VII.3.16) we ﬁnd

(x − y) =

∂

∂y

Ψ(x −y) −

2π

(λ|x − y|)e

−λ(x

−y

)

(x − y) = E

(x − y) = −

∂

∂y

Ψ(x − y)

(x − y) = −

∂

∂y

Ψ(x − y)

where

Ψ(x − y) =

4πλ



log |x − y| + K

(λ|x − y|)e

−λ(x

−y

)



From (VII.3.16) and the properties (VI I.3.10) of the f unction K

(z) near z =

0, we easily obtain, with r = |x − y|

=−

4π



log

2λr

− y

)(x

− y

)



+ o(1)

(x − y) −

4π

log

2λ

+ o(1), as λr → 0,

(VII.3.3 6)

where U is the Stokes fundamental tensor (IV.2.4). Using (VII.3.36) and

(VII.3.1 7), we then show the validity of (VII.3.22 ) also in two dimensions.

It shoul d be observed that, unlike the three-dimensional case (see (VII.3.21)),

relation (VII.3.36) for arbitrarily ﬁxed x, y becomes singular as λ → 0. This

is to be expected f or, as we are going to show, the tensor E vanishes at l arge

spatial distances, while U grows there logarithmically. Denote by ϕ the angle

made by a ray that starts from x and is directed toward y, with the direction

of the positive x

-axis. From (VII.3.1), (VII.3.15) and (VII.3.16) we derive in

the limit λr → ∞

440 VII Steady Oseen Fl ow in Exterior Domains

(x − y) = −

cos ϕ

4πλr

−s

√

2λπr



1 + cos ϕ −

1 − 3 cos ϕ

8λr

+ R(λr)



(x − y) = E

(x − y) =

sin ϕ

4πλr

−

−s

sin ϕ

√

2λπr



1 +

8λr

+ R(λr)



(x − y) =

cos ϕ

4πλr

−s

√

2π(λr)

3/2



s −

1 + 3 cos ϕ

+ R(λr)



(VII.3.3 7)

where

s = λr(1 −cos ϕ)

and the remainder R(t) satisﬁes

= O(t

−2−k

), as t → ∞ , k ≥ 0.

From (VII.3.37 ) it follows that unlike the Stokes fundamental tensor, which

grows logari thmically rapid for r → ∞, the Oseen fundamental tensor van-

ishes at large distances. It is this diﬀerence that al lowed Oseen to remove

the discrepancy generating the Stokes paradox. Relations (VII.3.37) produce

some useful estimates. We shall present them as a function of y, ﬁxing the

origin of coordinates at x = 0. Analogous estimates of E as a function of x

remain true if ϕ is replaced by π −ϕ, and the considerations m ade in Remark

VII.3.1 apply here.

From (VII.3.37)

it readily follows that E

(y) exhibits the same parabolic

“wake” region that was obtained in the three-dimensional case. In fact, if y is

interior to the parabola

|y|(1 − cos ϕ) = 1

we deduce the uniform bound

(y)| ≤

|y|

1/2

, as |y| → ∞, (VII.3.38)

while if

(1 − cos ϕ) ≥ |y|

−1+2σ

for some σ ∈ [0, 1/2] (VII.3.39 )

we have

(y)| ≤

|y|

1/2+σ

, as |y| → ∞. (VII.3.40)

However, unlike the three-dimensional case, the remaining components of E

do not present such a nonuniform behavior. In fact, observing that

Setting θ = π −ϕ, the parabolic region can be approximately described, for large

|y|, by

|θ| ≤ (2)

1/2

|y|

−1/2

VII.3 The Oseen Fundament al Solution 441

−2s

λr sin

ϕ = se

−2s

(1 + cos ϕ) ≤ e

−1

from (VII.3.37)

2,3

we recover the uniform estimates

(y)| ≤

|y|

, i = 1 , 2, as |y| → ∞. (VII.3.41)

Concerning the summability properties of E, denoting by A the exterior

of a unit circle, from (VII.3.41) we recover at once

(y) ∈ L

(A), for all q > 2, i = 1, 2. (VII.3.42)

Moreover, setting cos ϕ = ∓(y − λqr)/λqr, where “−” is taken if ϕ ∈ [0, π]

and “+” otherwise, by a direct calculation one shows for any q ≥ 1

2π

−qs

(1 + cos ϕ)

dϕ ≤

(4λqr)

2λqr

−y

(2λqr − y)

q−1/2

−1/2

≤

2(4λqr)

q−1/2

(2λqr)

∞

−y

−1/2

dy = c

1/2

with c

independent of r. As a consequence, from (VII.3.37)

we obtain

(y) ∈ L

(A), for all q > 3. (VII.3.43)

Likewise, setting

f(ϕ, r) ≡ −

cos ϕ

√

πλr

−s

√

2λπ



1 + cos ϕ −

1 − 3 cos ϕ

8λr



one has

2π

|f(ϕ, r)|

dϕ ≥

|f(ϕ, r)|

dϕ

(λqr)

2λqr



y − λqr

√

λπr

−y/q

√

2λπ



2λqr − y

−

3y −λqr

8λqr





−1/2

(2λqr −y)

1/2

dy,

and so, for all r suﬃciently large

2π

|f(ϕ, r)|

dϕ ≥

−y

(2λqr − y)

q−1/2

−1/2

≥

1/2

−y

−1/2

dy = c

1/2

442 VII Steady Oseen Fl ow in Exterior Domains

with c

independent of r, and from (VII.3 .37)

it follows that

(y) 6∈ L

(A), for all q ∈ [1, 3]. (VII.3.44)

So far as the behavior of the ﬁrst derivatives of E is concerned, diﬀerentiating

(VII.3.3 7) we derive the fo llowing bounds as |y| → ∞



∂E

(y)

∂y



≤

|y|



∂E

(y)

∂y



≤

|y|



∂E

(y)

∂y



≤

|y|

3/2



∂E

(y)

∂y



≤

|y|

, i = 1, 2.

(VII.3.4 5)

Sharper estimates can be obtained for the derivatives of E

and the deriva-

tive of E

with respect to y

whenever y is exterior to the “wake” region,

see Exercise VII.3.2. Mo reover, besides the obvious summability properties

obtainable from (VII.3.45) one can show (Exercise VII.3.2)

∂E(y)

∂y

∈ L

(A) for all q > 1

∂E(y)

∂y

∈ L

(A) for all q > 3/2

(VII.3.4 6)

while

∂E(y)

∂y

6∈ L

(A) for all q ∈ [1, 3/2]. (VII.3.47)

Further asymptotic bounds can be analogously derived for derivatives of

order higher than two. For instance, one shows the validity of the fol lowing

properties (see Exercise VII.3. 3):

E(y) ∈ L

(A), for all q > 1,

E(y)| ≤ c|y|

−(1+|α|) /2

, |α| ≥ 2, as |x| → ∞.

(VII.3.4 8)

Exercise VII.3.2 Let E be given as in ( VII.3.37). Show that for all σ ∈ [0, 1/2]

and all suﬃciently large |y|,

∂E

(y)

∂y

≤

|y|

1+2σ

∂E

(y)

∂y

∂E

(y)

∂y

≤

|y|

3/2+σ

in region (VII.3.39). Furthermore, show the validity of (VII.3.46) and (VII.3.47).

Exercise VII.3.3 Show estimate (VII.3.48).

VII.3 The Oseen Fundament al Solution 443

Exercise VII.3.4 Prove that (in two dimensions) the tensor E obeys the following

estimates for all R > 0:

∂B

|E(y)|

∂B

|∇E(y)| ≤ cR

−1/2

Hint: Use (VII.3.39) and (VII.3.41).

Exercise VII.3.5 Let E(x − y) ≡ E(x − y; 2λ) denote the Oseen tensor corre-

sponding to 2λ. Show the following homogeneity properties

E(x − y; 2λ) = 2λE(2λ(x − y); 1) for n = 3,

E(x − y; 2λ) = E(2λ(x − y); 1) for n = 2.

Remark VI I.3.2 Estimates a nalog ous to those presented for space dimen-

sion n = 2, 3 can be derived for al l n ≥ 4. Actually, from (VII.3.10)

and

(VII.3.1 1) it follows that the n-dimensional Oseen fundamental tensor E(x−y)

(deﬁned by (VII. 3.1), (VII.3.4), and (VII.3. 8)) sati sﬁes (VII.3.21) and, conse-

quently, becomes singular at x = y in such a way that (VII.3.22) is veriﬁed.

Furthermore, using the asymptotic expansion for large z:

(n−2)/2





1/2

−z

ν−1

k=0

Γ (n/2 + k −1/2)

k!Γ (n/2 − k −1/2)

(2z)

−k

+ σ

(z)





1/2

−z



1 +

4(n −2)

− 1

[4(n − 2)

−1] [4(n − 2)

− 3

]

2!(8z)

+ . . . + σ

(z)



with

= O(z

−k−ν

) as z → ∞, k ≥ 0,

see Watson (1962, p. 202 ), one can obtain estimates at large distances. For

instance, we can show

E(y)| ≤ c|y|

−(n−1+|α|)/2

, |y| → ∞, |α| ≥ 0,

E(y) ∈ L

(A), q >

n + 1

n − 1

∇E(y) ∈ L

(A), r >

n + 1

E(y) ∈ L

(A), s > 1.

(VII.3.4 9)



444 VII Steady Oseen Fl ow in Exterior Domains

In analogy with what we did for the Stokes problem, we now introduce

the Oseen volume potentials:

u(x) = 2λ

E(x − y) · F (y)dy

π(x) = −2λ

e(x − y) · F (y)dy,

(VII.3.5 0)

where F ∈ C

∞

). Since

E(x − y) · F (y)dy =

E(z) · F (x + z)dz,

one has u ∈ C

∞

) and, by the same token, π ∈ C

∞

). Moreover, it is

easy to show that u, π satisfy the Oseen system in R

. Actually, it is obvious

that ∇· u = 0. Also, using integration by parts and (VII.3.2) we deduce for

all x ∈ R

∆u(x) + 2λ

∂u(x)

∂x

−∇π = ∆(E ∗ F ) = 2λF (x).

Moreover, it is easy to show that the solution u, π behaves at large distances

exactly as the fundamental solution E, e. This immediately follows by ob-

serving that from (VII.3.49) we have

u(x) = 2λE(x) ·

F (y)dy + σ(x)

π(x) = −2λe(x) ·

F (y)dy + η(x)

(VII.3.5 1)

with

σ(x) = 2λ

(E(x −y) − E(x)) · F (y)dy

π(x) = −2λ

(e(x −y) − e(x)) · F (y)dy

and that, using the mean-value theorem and the assumption F ∈ C

∞

σ(x) = O(D

∇E(x))

η(x) = O(D

∇e(x))

|α| ≥ 0, |x| → ∞. (VII.3 .52)

Remark VI I.3.3 Starting with (VII.3.50)

and using Young’s theorem on

convolution (see Theorem II.11.1) one can prove at once L

-estimates for u, π

and their ﬁrst derivatives. This is due to the circumstance that, unlike the

Stokes tensor U , the Oseen tensor possesses global summability properties in

the whole of R

. For instance, from (VII.3.49) and (VII.3.21) we see tha t

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 445

E(y) ∈ L

) f or all q ∈ ((n + 1)/(n − 1), n) if n > 2,

E(y) ∈ L

) f or all q ∈ (3, ∞) if n = 2.

However, estima tes obtained in such a way would not be sharp and, therefore,

we shall not derive them here. Derivation of sharp estimates for the Oseen

potentials, by diﬀerent tools, will be the object of the next section. 

VII.4 Existence, Uniqueness, and L

-Esti mates in the

Whole Space

The objective of this section is to prove existence, uniqueness, and correspond-

ing estimates of solutions v, p to the nonhomogeneous O s een syst em

∆v + R

∂v

∂x

−∇p = Rf

∇ · v = g











in R

(VII.4.1 )

in homogeneous Sobolev spaces D

m,q

). These results, though sharing some

similarity with the analogous ones established for the Stokes system in Chap-

ter IV, wi ll diﬀer from these latter in some crucial f eatures that essentially

mirror the basic diﬀerences existing between the two fundamental tensors U

and E.

In establishing estimates for (VII.4.1), it is important to single out the

dependence of the constants entering the estimates on the dimensionless pa-

rameter R. We shall therefore consider the problem

∆v +

∂v

∂x

−∇p = f

∇ · v = g











in R

(VII.4.2 )

and establish corresponding estimates for i ts solutions. The analogous ones

for solutions to (VII.4.1) will then be obtained if we make the replacements

f → f/R

g → g/R

p → p/R

→ Rx

(VII.4.3 )

Unlike the corresponding estimates for the Stokes system, here we cannot

employ the Calder´on–Zygmund theorem because the kernel D

(x − y)

does not satisfy the assumption (II.11.15) of that theorem, that is,

−n

(ξ) 6= D

(αξ), α > 0.