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

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466 VII Steady Oseen Fl ow in Exterior Domains

v ∈ D

2,q

(Ω

) ∩ D

1,3q/(3−q)

(Ω

) ∩ L

3q/(3−2q)

(Ω)

∈ L

2q/(2−q)

(Ω) ∩ D

1,q

(Ω)

∂v

∂x

∈ L

(Ω)

p ∈ D

1,q

(Ω

(VII.5.1 3)

where p is the pressure ﬁeld associated to v by Lemma VII.1.1. Finally, the

following estimate holds:

kvk

2,Ω

+ |v|

1,2

+ R

2q/(2−q)

+ |v

1,q



∂v

∂x



+bkvk

3q/(3−2q)

+ R

1/3

|v|

1,3q/(3−q),Ω

R + |v|

2,q,Ω

R + |p|

1,q,Ω

≤ c



R(kfk

+ (1 + R)|f |

−1,2

) + (1 + R)

∗

1/2,2(∂Ω)



(VII.5.1 4)

where b = min{1, R

2/3

} and c = c(q, Ω, R).

Proof. Uniqueness is already known from Theorem VII.1.2. Concerning exis-

tence, we proceed as follows. We ta ke ε = 1/m, m ∈ N, in (VII.5 .1) and denote

by v

, p

the corresponding generalized solution and the associated pressure

ﬁeld which, by Lemmas 5.2 and 5.3 exist and satisfy inequalities (VII.5.9 )–

(VII.5.1 1) with constants c

, c

and c independent of m. In particular, such

inequalities for any ﬁxed R > δ(Ω

) lead to the uniform bo und

2q/(2−q)

+ |v

1,q



∂v

∂x



+ |v

1,3q/(3−q),Ω

+kv

3q/(3−2q)

+ |v

1,2

+ |v|

2,q,Ω

R + |p

1,q

≤ M

(VII.5.1 5)

with M independent of m. Using the weak compactness of the spaces L

and

m,r

, 1 < r < ∞ (Theorem II.2.4 and Theorem II.3.1 and Exercise II. 6.2),

together with the strong compactness results of Exercise I I.5.8, from (VII.5 .9),

(VII.5.1 1) and (VII.5.15) we then deduce the existence of a subsequence, de-

noted again by {v

, p

}, and of ﬁelds v, p verifying (VII.5.13) and (VII.5.14)

and, moreover, as m → ∞,

→ v, weakly in W

1,2

(Ω

) and strongly in L

(Ω

), (VII.5.16)

for any R > δ(Ω

). It is simple to show tha t v satisﬁes (VII. 1.1) for all

ϕ ∈ D(Ω). To see this, we notice that v

satisﬁes (VII.5 .8) with ε = 1/m,

which in view of (VII.5.16) reduces to (VII.1.1) in the l imit m → ∞. Clearly,

v is weakly divergence-free and, by (VII.5.1 6) and Theorem II.4.1, v = v

∗

∂Ω in the trace sense. To prove the theorem compl etely, it remains to show

condition (i) of Deﬁnition VII.1.1. Actually, we shall prove something more;

that is,

VII.6 Representation of Solutions. Asymptotics and Related Results 467

lim

|x|→∞

v(x) = 0. (VII. 5.17)

In fact, from the property v ∈ D

2,q

(Ω

), 1 < q < 3/2, and from Theorem

II.6.1 we obtain v ∈ D

1,2q/(2−q)

(Ω

). Thus

v ∈ D

1,2q/(2−q)

(Ω

) ∩L

3q/(3−2q)

(Ω

)

and (VII.5.17) follows from Theorem II.9.1. ut

VII.6 Representation of Solutions. Behavior at Large

Distances and Related Resul ts

We shall presently investigate the behavior at inﬁnity of solutions to the Oseen

system and, in particular, we shall determine its asymptotic structure. To

reach this goal, we will pattern essentially the same ideas and techniques used

for a nalog ous questions within the Stokes approximation in Section IV.8 and

Section V.3 and therefore, here and there, we may leave details to the reader.

Let us begin to show a representation formula for smooth solutions i n a

bounded domain of cla ss C

. Denoting, as usual, by T the stress tensor of a

given ﬂow, for v, p and u, π enough regular ﬁelds the following identities hold:

Ω



∇ · T (v, p) + R

∂v

∂x



· u = −

Ω



T ( v, p) : ∇u + Rv ·

∂u

∂x



∂Ω



u ·T (v, p) · n + Rv · u e

·n



Ω



∇ · T (u, π) + R

∂u

∂x



· v = −

Ω



T (u, π) : ∇v + Rv ·

∂u

∂x



∂Ω

(v · T (u, π) · n) ,

(VII.6.1 )

where, as usual, e

denotes the unit vector along the posi tive x

-axis. Assum-

ing u and v solenoidal impli es

Ω

T (u, π) : ∇v =

Ω

T ( v, p) : ∇u

and so, subtracting (VII.6.1)

from (VII .6.1)

, we obtain

Ω



∇ · T (v, p) + R

∂v

∂x



·u −



∇ · T (u, π) + R

∂u

∂x



· v



∂Ω

(u · T (v, p) −v ·T (u, π) + Rv · u e

) ·n.

(VII.6.2 )

468 VII Steady Oseen Fl ow in Exterior Domains

Identity (VII.6.2) is the Green’s formula for the Oseen system. Proceeding a s

in Section IV.8, i t is easy to derive from (VII.6.2) a representation formula

for v and p sati sfy ing the Oseen system. Actually, for ﬁxed j = 1, . . ., n and

x ∈ Ω we choose

u(y) = w

(x − y) ≡ (E

, E

, . . . , E

)

π(y) = e

(x − y),

(VII.6.3 )

where E, e is the fundamental sol ution introduced i n Section VII.3. Replacing

(VII.6.3 ) into (VII.6.2) with Ω

≡ Ω − {|x − y| ≤ ε} i n place of Ω and then

letting ε → 0, i n view of (VII.3.18) and (VII.3.22) we recover

(x) = R

Ω

(x − y)f

(y)dy +

∂Ω

(y)T

, e

)(x − y)

−E

(x − y)T

(v, p)(y) − Rv

(y)E

(x − y)δ

dσ

(VII.6.4 )

where

Rf = ∆v + R

∂v

∂x

−∇p. (VII.6.5)

We now turn to the representation for the pressure. By means of classical

potential theory one can show that if f is H¨older continuous, the volume

potentials

(x) = R

Ω

(x − y)f

(y)dy

S(x) = −R

Ω

(x − y)f

(y)dy

are (at least) of class C

and C

, respectively, and, moreover,

∂S

∂x

+ Rf

in Ω, (VII.6.6)

where

Lu ≡ ∆u + R

∂u

∂x

From (VII. 6.4) and (VII.6.5) we have

∂p

∂x

+ Rf

= Lv

= LW

∂Ω

, e

)

−(LE

(v, p) − Rv

(VII.6.7 )

and observing that e

is ha rmonic (for x 6= y) and that ∂e

/∂x

= ∂e

/∂x

from (VII.3.19) and from the deﬁnition of T it also follows that

VII.6 Representation of Solutions. Asymptotics and Related Results 469

, e

) = Le

∂

∂x

(LE

) +

∂

∂x

(LE

)

= −R

∂e

∂x

−2

∂

∂x

(VII.6.8 )

Using (VII.6.5) and (VII.6.8) in (VII.6 .7) we conclude the validity (up to a

constant) of the following formula for all x ∈ Ω:

p(x) = −R

Ω

(x − y)f

(y)dy +

∂Ω



(x − y)T

(v, p)(y)

−2v

(y)

∂

∂x

(x − y) − R[e

(x − y)v

(y)

−v

(y)e

(x − y)δ

]



dσ

(VII.6.9 )

Formulas (VII.6. 4) and (VII.6.9) can b e generalized toward the following

two directions:

(i) To derive analogous formulas fo r derivatives of v and p of arbitrary o rder;

(ii) To show their validity with v and p only belo nging to suitable Sobolev

spaces.

The ﬁrst issue is trivially a chieved (provided v, p, and f are suﬃciently

smooth) by replacing in (VII.6.4) and (VII.6.9) v, p and f with D

v, D

and D

f , respectively. The second o ne can be proved along the same lines

of Theorem IV.8.1. However, we need the following result, whose validity is

established by means of Theorem IV.4.1 and Theorem IV.5.1.

Lemma VII.6.1 Let Ω be a bounded domain of class C

m+2

, m ≥ 0. Fo r any

f ∈ W

m,q

(Ω), v

∗

∈ W

m+2−1/q,q

(∂Ω), 1 < q < ∞, with

∂Ω

∗

· n = 0,

there exists one and only one function v ∈ W

m+2,q

(Ω), p ∈ W

m+1,q

(Ω)

satisfying a.e. the Oseen problem

∆v + R

∂v

∂x

= ∇p + Rf

∇ · v = 0











in Ω

v = v

∗

at ∂Ω.

(VII.6.1 0)

Proof. The existence of a generalized solution is at once established with the

Galerkin method used in the proof of Theorem VII.2. 1. Employing Theorem

IV.4. 1 and Theorem IV.5.1 we then show that such a solution satisﬁes all

requirements stated in the theorem. Uniqueness is a simple exercise. ut

470 VII Steady Oseen Fl ow in Exterior Domains

Lemma VII.6.1, together with an argument entirely analogous to that o f

Theorem IV.8.1, implies the following result.

Lemma VII.6.2 Let Ω sati sfy the assumption of Lemma VII.6.1 . Let v ∈

m+2,q

(Ω), p ∈ W

m+1,q

(Ω) be a solution to (VII.6.10)

corresponding to

f ∈ W

m,q

(Ω), m ≥ 0, 1 < q < ∞. Then, for all | α| ∈ [0, m] and almost all

x ∈ Ω,

(x)= R

Ω

(x − y)D

(y)dy +

∂Ω

(y)T

, e

)(x − y)

−E

(x − y)T

v, D

p)(y)−RD

(y)E

(x−y)δ

dσ

p(x) =−R

Ω

(x − y)D

(y)dy +

∂Ω



(x − y)T

v, D

p)(y)

−2D

(y)

∂

∂x

(x − y) − R[e

(x − y)D

(y)

−D

(y)e

(x − y)δ

]



dσ

Our next task is to extend the above results to the case when Ω i s an ex-

terior domain. As in the case o f the Stokes approximation, we shall use a suit-

able “truncation” of the Oseen fundamental tensor, along the lines suggested

by Fujita (1961) in the nonlinear context. Thus, the Oseen-Fujita truncated

fundamental solution E

(R)

, e

(R)

is deﬁned by (VII.3.1), (VII.3.4), (VII.3.8),

and (VII.3.11) with Φ replaced by ψ

Φ, where ψ

is the “cut-oﬀ” function

introduced in Section V.3. Clearly,

(R)

(x − y) = E

(x − y), e

(R)

(x − y) = e

(x − y), if |x −y| ≤ R/2,

while E

(R)

(x −y) and e

(R)

(x −y) va nish identically for |x −y| ≥ R. Further-

more, from (VII.3.1) and (VII.3.2) one immediately obtains



∆ − R

∂

∂y



(R)

(x − y) −

∂

∂y

(R)

(x −y) = H

(R)

(x − y)

∂

∂y

(R)

(x −y) = 0,

for x 6= y

(VII.6.1 1)

where H

(R)

(x − y) is deﬁned by H

(R)

(0) = 0 and

(R)

(x − y) = δ

∆



∆ − R

∂

∂y



(ψ

Φ).

Similar to the function H

(R)

(x−y) introduced for the Stokes-Fujita truncated

fundamental solution, H

(R)

(x − y) is also an inﬁnitely diﬀerentiable function

VII.6 Representation of Solutions. Asymptotics and Related Results 471

vanishing unless R/2 < |x − y| < R. However, due to the inhomogeneity of

the Oseen diﬀerential operator, the unif orm asymptotic properties of H

(R)

as R → ∞ are somewhat diﬀerent from those of H

(R)

. In fact, we have the

following estimate, whose proof is left to the reader

(R)

(x − y)| = O(R

−(n+1+|α|)/2

), |α| ≥ 0. (VII.6.12)

The next result is proved exactly as in Lemm a V.3.1.

Lemma VII.6.3 Let Ω be an arbitrary domain of R

. Let v ∈ W

1,2

loc

(Ω)

be weakly divergence-free and satisfy (VII.1.1) for all ϕ ∈ D(Ω). Then, if

f ∈ W

m,q

loc

(Ω) it follows that v ∈ W

m+2,q

loc

(Ω) and, moreover, for all ﬁxed

d > 0 and all |α| ∈ [0, m], v obeys the identity

(x) =

(x)

(d)

(x − y)D

(y)dy −

β(x)

(d)

(x − y)D

(y)dy

(VII.6.1 3)

for a lmost all x ∈ Ω with dist (x, ∂Ω) > d, where β(x) = B

(x) − B

d/2

(x).

Lemma VII.6.3 allows us to argue as in Theorem V.3.1, to show the fol-

lowing.

Theorem VII.6.1 Let v be a q-generalized solution to the Oseen problem

in an exterior domain Ω with v ∈ L

(Ω

), f or some s ∈ (1, ∞) and some

R > δ(Ω

). Then, if f ∈ W

m,r

(Ω), m ≥ 0, n/2 < r < ∞,

it follows that

lim

|x|→∞

v(x) = 0, 0 ≤ |α| ≤ m .

To conclude this section it remains to investigate the structure of the

solution at inﬁnity and the corresponding rate of decay. Again, as in the Stokes

approximation, we shal l use the truncated fundamental solution. Starting with

the Green’s identity (VII.6.2) a nd choosing as u, π this latter solution we may

readily show, by means of the same procedure adopted in Chapter V, the

validity o f the identities

(x) = R

Ω

(x − y)f

(y)dy +

∂Ω

(y)T

, e

)(x − y)

−E

(x − y)T

(v, p)(y) − Rv

(y)E

(x − y)δ

dσ

−

Ω

(R)

(x − y)v

(y)dy

(VII.6.1 4)

Bounds more accurate than those given in (VII.6.12) can be obtained according

to whether we are inside or outside the “wake” region. However, (VII.6.12) will

suﬃce for our purposes.

See footnote 1 of Section V.3.

472 VII Steady Oseen Fl ow in Exterior Domains

and

∂

∂x

p(x) =

∂

∂x



−R

Ω

(x − y)f

(y)dy +

∂Ω

(x − y)T

(v, p)(y)

−2v

(y)

∂

∂x

(x − y)

−R[e

(x − y)v

(y) − v

(y)e

(x − y)δ

]}n

dσ



−

Ω



∆ − R

∂

∂x



(R)

(x − y)v

(y)dy.

(VII.6.1 5)

Relations (VII. 6.14) and (VII.6.1 5), which hold for almost all x ∈ Ω, are valid

for Ω of class C

, v ∈ W

2,q

loc

(Ω), q ∈ (1, ∞), and f belongi ng to L

(Ω) and

with compa ct support in Ω. Moreover, R is so large that B

(x) contains Ω

and the support of f .

Denote by I(x) the last i ntegral on the right-hand side

of (VII.6.14). It is easy to show that if

Ω

R/2,R

|v| = O(R

I(x) is a polynomial whose degree depends on k and n. In fact, observing that

I(x) is independent of R, we have

I(x) = −

Ω

(R)

(x − y)v

(y)dy

and so, by (VII.6.12),

I(x)| ≤ cR

−(n+1+|α|)/2

Ω

R/2,R

|v| ≤ c

−(n+1+|α|−2k)/2

Thus, choosing |α| = 2k − n (say), we deduce D

I(x) = 0. Evidently, since

as |x| → ∞

I(x) = v(x) + o(1),

I(x) must reduce to a constant whenever v does not “grow” too fast at la rge

distances. Also, if I(x) is a constant, the last integral on the right-hand side

of (VII. 6.15) is identically zero. Bearing this in mind, reasoning in complete

analogy with Theorem V.3.2 and recalling the estimate for the Oseen funda-

mental solution given in Section VII.3, we obtain

Theorem VII.6.2 Let Ω be a C

-smooth, exterior do mai n and let v ∈

2,q

loc

(Ω), q ∈ (1, ∞), be weakly divergence-free and satisfy (VII.1.1) for all

ϕ ∈ D(Ω) with f ∈ L

(Ω). Assume further that the supp ort of f is bounded.

Then, if at least one o f the following conditions is satisﬁed as |x| → ∞:

Recall that in B

(x) the fundamental solution and the truncated fundamental

solution coincide

VII.6 Representation of Solutions. Asymptotics and Related Results 473

(i)

|v(x)| = o(|x|)

(ii)

|x|≤r

|v(x)|

(1 + |x|)

n+t

dx = o(log r), some t ∈ (1, ∞),

there exist vector and scalar constants v

, p

such that for almost all x ∈ Ω

we have

(x) = v

+ R

Ω

(x − y)f

(y)dy +

∂Ω

(y)T

, e

)(x − y)

−E

(x − y)T

(v, p)(y) − Rv

(y)E

(x − y)δ

dσ

≡ v

+ v

(1)

(x),

(VII.6.1 6)

and

p(x) = p

− R

Ω

(x − y)f

(y)dy +

∂Ω

(x − y)T

(v, p)(y)

−2v

(y)

∂

∂x

(x − y)

−R[e

(x − y)v

(y) − v

(y)e

(x − y)δ

]}n

dσ

≡ p

+ p

(1)

(x).

(VII.6.1 7)

Moreover, as |x| → ∞, v

(1)

(x) and p

(1)

(x) are inﬁnitely diﬀerentiable and

there the following asymptotic representations hold:

(1)

(x) = E

(x)M

+ σ

(x)

(1)

(x) = −e

(x)M

∗

+ η(x),

(VII.6.1 8)

where

= −

∂Ω

(v, p) + Rδ

+ R

Ω

∗

= −

∂Ω

(v, p) + R[δ

− δ

]}n

+ R

Ω

(VII.6.1 9)

and, for all |α| ≥ 0,

σ(x) = O(|x|

−(n+|α|)/2

)

η(x) = O(|x|

−n−|α|

(VII.6.2 0)

Remark VI I.6.1 Theorem VI I.6.2 asserts, among other thing s, that every q-

weak solution to (VII.0.2) and (VII.0 .3) behaves asymptotically as the Oseen

474 VII Steady Oseen Fl ow in Exterior Domains

fundamental solution. In particular, taking into account the properties of this

solution, every q-weak solution exhibits a paraboloidal wake region in the

direction of the positive x

-axis; see Remark VII.3.1. 

Some interesting consequences of Theorem VII.6.2 are l eft to the reader

in the following exercises.

Exercise VII.6.1 Let v satisfy the assumption of Theorem VII.6.2. Show that, for

all suﬃciently large R,

∂B

+ ∇v : ∇v

≤ c

−(n−1)/2

∂B

|p − p

≤ c

−(n−1)

Hint: Use Theorem VII.6.2 together with Exercise VII.3.1 and Exercise VII.3.4.

Exercise VII.6.2 The following result generalizes uniqueness Theorem VII.1.2.

Let v, p be a q-generalized solution to the Oseen problem (VII.0.3), (VII.0.2) in an

exterior domain Ω of class C

. Show that if f ≡ v

∗

≡ 0 then v ≡ 0, p ≡const.

Under these latter assumptions on the data, show that if v, p is a corresponding

smooth solution with v = o(1) as |x| → ∞, then v ≡ 0, p ≡ const.

Exercise VII.6.3 Show that the remainder σ in (VII.6.18)

has the following

summability properties:

σ ∈ L

(Ω

), for all q > n/(n − 1)

σ ∈ L

(Ω

), for all q > (n + 1)/n, if Φ ≡

∂Ω

∗

· n = 0

R > δ(Ω

Hint: As |x| → ∞, it is σ(x) = O(e(x)) if Φ 6= 0, and O( ∇E(x)) if Φ = 0 . Then use

the summability prop erties of e and ∇E.

Exercise VII.6.4 Let v, p be a smooth solution to the Oseen problem in an exterior

domain Ω ⊆ R

, n = 2, 3, with v = 0 at ∂Ω. Show that if v 6≡ 0, necessarily

kv − v

∞

2,Ω

= ∞, for any choice of v

∞

∈ R

Hint: Recall that E 6∈ L

(Ω

Exercise VII.6.5 (Olmstead & Gautesen 1968) Show the following “drag paradox”

for the Oseen approximation. Let B be a (smooth) body moving in a viscous liquid

that ﬁlls the whole space, with no spin and translational velocity λe

. Set Ω :=

− B, and denote by D(e

) := e

∂Ω

n · T , the drag exerted by the liquid on B

in t he Oseen approximation, where n i s the inner unit normal to ∂Ω (≡ ∂B). Prove

that t he drag is the same if the direction of the translational velocity is reversed,

namely, D(e

) = −D(−e

). Hi nt: Use (appropriately!) (VII.6.1) on the domain Ω

then let R → ∞ and employ the asymptotic properties of Theorem VII.6.2.

The representation formula (VII.6.1 6) allows us to obtain an interesting

asymptotic estimate for the vortici ty ﬁeld ω = ∇×v in three dimensions. To

this end, we observe that, setting

If n ≥ 4 this statement no longer holds since E ∈ L

(Ω

VII.7 Existence, Uniqueness, and L

-Estimates in Exterior Domains 475

f(x − y) ≡

−ρ

4πR| x −y|

, ρ =

(|x − y| − (x

−y

)), (VII.6.21)

by a direct calculation one shows

∇

×(E(x − y) · G(y)) = ∇

f(x −y) × G(y)

and, consequently, (VII.6.16) furnishes, for all suﬃciently large |x|

ω(x) = R

Ω

∇

f(x − y) ×f (y)dy +

∂Ω

[∇

(∇

f(x − y) · n) × v(y)

+∇

f(x − y) ×(−T (v, p)(y) · n − Rv(y) · n)]dσ

(VII.6.2 2)

Applying the m ean value theorem in the integrands in (VII.6.22), we easily

deduce

ω(x) = ∇f (x) ×M + O(D

f(x)), as |x| → ∞, (VII.6.23)

where the vector M is deﬁned in (VII.6 .19)

. From (VII.6.21) and (VII.6.2 3) it

is apparent that in the region R situated outside the wake region (VII.3.27) and

suﬃciently far from ∂Ω, the vorticity decays exponentially fast. Thi s means,

essentially, that the ﬂow is potential in R, as expected from the physical point

of view. The reader will prove with no diﬃculty that an analogous conclusion

holds in the case of a plane ﬂow wi th ω = ∂v

/∂x

−∂v

/∂x

; see Clark (1971,

§§2.2 and 3.2).

VII.7 Existence, Uniqueness, and L

-Esti mates in

Exterior Domains

The aim of this section is to investigate to what extent the theorems proved in

Section VII.4 in the whole space can be extended to the more g eneral situation

when the region of ﬂow is an exterior domain. Of course, the results we shall

prove rely heavily on those of Section VII.4 and, like those, they wi ll be similar

to those derived for the Stokes problem in Chapter V; nevertheless, they will

diﬀer from these in some crucial features that resemble the diﬀerence existing

between the fundamental tensors U and E.

Let us begin to consider the Oseen problem (VII.0.2), (VII.0.3) in an (ex-

terior) domain Ω of class C

m+2

, m ≥ 0, with data

f ∈ C

∞

(Ω), v

∗

∈ W

m+2−1/q

(∂Ω), 1 < q < ∞.

By Theorem VII.2.1 and Theorem VII.5.1 we may then construct a solution

v, p such that

v ∈ W

m+2,q

loc

(Ω) ∩ C

∞

(Ω), p ∈ W

m+1,q

loc

(Ω) ∩ C

∞

(Ω)