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

VII

Steady Oseen Flow in Exterior Domains

e vidi le ﬁammelle andare avante,

lasciando retro a s`e l’aer dipinto.

DANTE, Purgatorio XXIX, vv. 73-74

Introduction

As we emphasized in the Introduction to Chapter V, the Stokes approximation

may fail to describe the physical properties of a system constituted by an

object B m oving by assigned rigid motion with “sm all” translational (v

) and

angula r (ω) velocities in a viscous liquid, at least at “large” distances from

B, where the viscous eﬀects become less important.

In particular, for B a ball translating without rotating (that is, ω = 0),

the explicit solution one ﬁnds (see (V.0. 4)) exhibits no wake behind the body

and is, therefore, unacceptable from the physical viewpoint. Moreover, for B a

circle (plane motion), the analogous problem admits no solution except for the

trivial one, thus leading to the Stokes paradox. It is interesting to remark that

a sort o f similar paradox also arises in the three-dimensional case, the moment

one tries to evaluate the ﬁrst-order (in the Reynolds number) correction to

the zero-th order solution (V.0.4); see Whitehead (1888). In addition to all

the above, as observed by Oseen (1927, p.165 ), for the solution (V.0.4) we

obtain, after a simple calculation,



v · ∇v

∆v



→ ∞ as |x| → ∞,

no matter how small |v

| is, thus violating the assumption under which the

Stokes equati ons are derived (see the Introduction to Chapter IV).

417G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,

DOI 10.1007/978-0-387-09620-9_7, © Springer Science+Business Media, LLC 2011

418 VII Steady Oseen Fl ow in Exterior Domains

As we a lready remarked, it is reasonable to argue that these “anomalous

behaviors” must be chieﬂy ascribed to the fact that the Stokes approximation

completely disregards the inertia of the liquid or, in equivalent mathematical

terms, it ignores possibly signiﬁcant information arising from the nonlinear

term in the Navier-Stokes equation (I.0.1 )

. One is thus natural ly lead to

introduce other linearizations of (I.0.1)

that may, somehow, take into account

this feature.

With this in mind, C.W. Oseen proposed in 1910 (see also Oseen 1927 ,

§15) a linearization of the Navier-Stokes equations with the main objective of

avoiding the paradoxes and the incongruities related to the Stokes approxima-

tion.

The original equations introduced by Oseen (which we will refer to as

Oseen approximation) are f ormally obtained by linearizing the Navier-Stokes

equations around a nonzero purely translational moti on v = v

, p = p

where v

and p

are given constant vector and scala r quantities, respectively.

However, for reasons that are mai nly dictated by a considerable number of

signiﬁcant applications (see Gal di 2 002, and the references therein), we shall

analyze a more general approximation (which we wil l refer to as generalized

Oseen approximation), consisting of linearizing (I.2.2)

around the (nonzero)

rigid motion, v = v

+ ω ×x ≡ V (x), p = p

, where v

and ω are prescribed

(constant) vectors and p

is a given scalar quantity. We recall that, from a

physical viewpoint, v

and ω represent the (constant) translational and the

angula r velocity of the body B, respectively, when the motion of the liquid is

referred to a frame attached to B.

Thus, denoting by Ω the exterior region occupied by the liqui d, from

(I.2. 3)

we obtain the following generalized O s een syst em

ν∆v + V · ∇v − ω ×v = ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω,

(VII.0.1 )

where v

∗

is a prescribed ﬁeld at the boundary wall. To (VII.0.1) we append

the condition at inﬁnity

lim

|x|→∞

v(x) = 0 . (VII.0.2)

As kindly pointed out to me by Professor Josef Bemelmans, an independent anal-

ysis of these questions, mostly motivated by t he study of the range of validity of

Stokes formula for the drag, was perf ormed by Fritz Noether (1911).

See also the introductions to Chapter X and Chapter XI.

Sometimes, t he system (VII.0.1) with V = 0 is also referred to as Sobolev system;

see, e.g., Maslennikova (1973).

The Oseen approximation is typical for a ﬂow occurring in an exterior region.

In a bounded region it loses its physical meaning, while, from the mathematical

point of view, it presents no diﬃculties and can be handled as a corollary to

the theory developed for the Stokes problem i n Section IV.4–Section IV.6; see

Theorem VII.1.1.

VII Introduction 419

In the current chapter we begin to investigate the properties of solutio ns

to problem (VII.0.1)–(VII. 0.2) in the simpler case when ω = 0, that is, to the

problem originally formulated by Oseen, whereas in the next chapter we shall

study it in its full generality, namely, with both v

and ω being non-zero.

It should be stressed that results from the original Oseen approximation

have long been recognized to be much more successful than that of Stokes. As

a matter o f fact, at least in the particular case of the translational motion of

a ball into a liquid, Oseen found a paraboloidal wake region behind the body

(Oseen 1 910, 1927 §16; Goldstein 1929). Furthermore, in the two-dimensional

analogue, i.e., an inﬁnite circular cylinder moving steadily in a viscous liquid,

Lamb (1911 ) ﬁrst proved the existence of a soluti on to (VII.0.1 ), (VII.0.2)

with V ≡ v

6= 0, that exhibit a wake region, thus removing the paradox

coming from the Stokes approximation.

The aim of this chapter is to i nvestigate existence, uniqueness, and the

validity of corresponding estimates in homogeneous Sobolev spaces D

m,q

for

solutions to (VII.0.1), (VII.0.2) with V ≡ v

6= 0, in an arbitrary exterior

domain Ω. All main ideas are taken from Galdi (1992).

The lines we shall follow are essentially the same we followed in Chapter V

for the exterior Stokes problem, even though the study is here somehow com-

plicated by the more involved form of the fundamental soluti on to (VII.0.1 )

1,2

in the whole space R

. However, because of the diﬀerent structure of the equa-

tions, the results we shall obtain are substantially diﬀerent from those proved

for the Stokes problem. In this respect, we will show that problem (VII.0.1),

(VII.0.2 ) (with V ≡ v

6= 0 and with suﬃciently smooth data) is solvable in

three dimensions and two di mensions a nd that, if f is of bounded support, the

corresponding solutions exhibit a paraboloidal “wake region” in a direction

opposite to v

. This fact implies, in particular, that for problem (VII.0.1),

(VII.0.2 ) with V ≡ v

6= 0, no “Stokes paradox” arises and that the Os-

een approximation is, in this respect, better than that proposed by Stokes.

Also, as in the Stokes problem, the existence of q-generalized (in D

1,q

) and

“strong” solutions (in D

m,q

, m > 1) is proved onl y for q in a certain range

depending on the space dimension n; however, we ﬁnd that R

is larger

than the analogous range R

for the Stokes problem. Precisely, we show that,

formally, R

= R

n+1

. This circumstance will lead to important consequences

in the nonlinear context, when treating the motion of an object translating

with constant velocity into a viscous liquid; see Chapters X, and XII.

Finally, we shall consider the behavior of solutions to (VII. 0.1), (VII.0.2)

with V ≡ v

in the limit of vanishing v

, with special emphasis on the case of

plane m otion. In this la tter circumstance, we ﬁnd that such solutions tend to

those of the analogous Stokes syst em, i.e., (VII.0.1) with v

= 0 . However, as

It should be observed, however, that the Oseen approximation leads to other

paradoxical consequences in disagreement with the actual slow motion of a body

into a viscous li quid; see Filon (1928), Imai (1951), Olmstead & Hector (1966),

Olmstead & Gautesen (1968), and Olmstead (1968); see also Exerci se VII.6.5.

420 VII Steady Oseen Fl ow in Exterior Domains

expected in view of the Stokes paradox, the l imiting process does not preserve

condition (VII.0.2), which is, in fact, satisﬁed if and only if the data obey the

compatibility condition determined in Section V.8.

For later purposes, we shall ﬁnd it convenient to put (VII.0.1), (VII.0.2 )

with V ≡ v

into a suitable dimensionless form, and so we need comparison

length d a nd velocity U. Without loss, we set v

= v

, v

> 0, and take

U = v

. Moreover, i f |Ω

| 6= ∅, we can take d = δ(Ω

), and so, introducing

the Reynolds number

R =

system (VII.0.1) becomes

∆v + R

∂v

∂x

= ∇p + Rf

∇ · v = 0











in Ω

v = v

∗

at ∂Ω,

(VII.0.3 )

where v, v

∗

, p and f are now nondimensional quantities. If Ω ≡ R

the above

choice of d is no longer po ssible, even though we can still give a meaning to

(VII.0.3 ), which is what we shall do hereafter.

VII.1 Gene ralized Solutions. Regularity and Uniqueness

In analogy with similar questions treated for the Stokes approximation, we

shall begin to give a generalized formulation of the Oseen problem. To this

end, let us multiply (VII.0.3)

by ϕ ∈ D(Ω) a nd integrate by parts to obtain

formally

(∇v, ∇ϕ) − R(

∂v

∂x

, ϕ) = −R[f, ϕ]. (VII.1.1)

Deﬁnition VII.1.1. A vector ﬁeld v : Ω → R

is call ed a q-weak (or q-

generalized) solution to (VII.0.2), (VII.0.3) if for some q ∈ (1, ∞)

(i) v ∈ D

1,q

(Ω);

(ii) v is (weakly) divergence-free in Ω;

(iii) v assumes the value v

∗

at ∂Ω (in the trace sense) o r, if the veloci ty at

the boundary is zero, v ∈ D

1,q

(Ω);

(iv) lim

|x|→∞

n−1

|v(x)| = 0;

(v) v veriﬁes (VII.1.1) for all ϕ ∈ D(Ω).

If q = 2, v will be simply cal led a weak (or generalized) solution to (VII.0. 2),

(VII.0.3 ).

VII.1 Generalized Solutions. Regularity and Uniqueness 421

Remark VI I.1.1 If v is a q-weak solution, then, by Lemma II.6.1, we have

that v ∈ W

1,q

loc

(Ω), and, if Ω is locally Lipschitz, v ∈ W

1,q

loc

(Ω). Regarding (iii),

see Remark V.1.1. 

If the function f has some mild degree of regularity, to each q-weak solution

we can associate a corresponding pressure ﬁeld in the usual way. Speciﬁcally,

we have the following lemma whose proof, being entirely analogous to that of

Lemma IV.1.1, will be omitted.

Lemma VII.1.1 Let Ω be an exterior domain in R

, n ≥ 2. Suppose f ∈

−1,q

(Ω

), 1 < q < ∞, fo r any bounded subdomain Ω

, with Ω

⊂ Ω. Then,

to every q-weak solution v we can associate a pressure ﬁeld p ∈ L

loc

(Ω) such

that

(∇v, ∇ψ) − R(

∂v

∂x

, ψ) = (p, ∇ ·ψ) − R[f , ψ] (VII.1.2)

for all ψ ∈ C

∞

(Ω). Furthermore, if Ω is locally Lipschitz and f ∈

−1,q

(Ω

), R > δ(Ω

), then p ∈ L

(Ω

Remark VI I.1.2 The last result stated in Lemma VI I .1.1 is weaker than

the analogous o ne proved for the Stokes problem in Lemma V.1.1, where, for

Ω locally Lipschitz, one has p ∈ L

(Ω) whenever f ∈ D

−1,q

(Ω). This is due

to the fact that, in the case at hand, the functional

(∇v, ∇ψ) − R(

∂v

∂x

, ψ) + R[f , ψ]

is not continuous in ψ ∈ D

1,q

(Ω) if v ∈ D

1,q

(Ω) only, because a priori we

can not ﬁnd a constant c = c(v) such that

∂v

∂x

, ψ)| ≤ c |ψ|

1,q

, for all ψ ∈ C

∞

(Ω) .

Consequently, we cannot apply Corollary III.5.1 but only the weaker version,

Corollary III.5.2. Notice, however, that, by the very deﬁnition of q-weak so-

lution, if f ∈ D

−1,q

(Ω), then we can ﬁnd C > 0 such that

∂v

∂x

, ϕ)| ≤ C |ϕ|

1,q

, for all ϕ ∈ D(Ω) .

In any case, by using a completely diﬀerent approach, in Section VII.7 (see

Theorem VII.7.2), we shall show that if the region of motio n i s of class C

and the exponent q ranges in the interval (n/(n −1), n + 1), the pressure ﬁeld

p belongs to L

(Ω), provided, of course, that f ∈ D

−1,q

(Ω). Furthermore, in

Theorem VII.7.3 it will be proved that the same property continues to hold

for q ≥ n + 1. It seems therefore an open question to ascertain whether or not

for q-weak soluti ons with q in the interval (1, n/(n − 1)] the corresponding

pressure p has a suitable degree of summability at large distances. 

422 VII Steady Oseen Fl ow in Exterior Domains

The next result establi shes the regularity of q-weak solutions.

Theorem VII.1.1 Let f ∈ W

m,q

loc

(Ω), m ≥ 0, 1 < q < ∞, and let

v ∈ W

1,q

loc

(Ω), p ∈ L

loc

(Ω),

with v weakly divergence-free, satisfy (VII.1 .2) for all ψ ∈ C

∞

(Ω). Then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω).

In particular, if f ∈ C

∞

(Ω), then v, p ∈ C

∞

(Ω). Furthermore, if Ω is o f class

m+2

and

f ∈ W

m,q

loc

(Ω), v

∗

∈ W

m+2−1/q,q

(∂Ω),

then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω) ,

provided v ∈ W

1,q

loc

(Ω).

In particular, if Ω is of class C

∞

and f ∈ C

∞

(Ω),

∗

∈ C

∞

(∂Ω) then v, p ∈ C

∞

(Ω

), for all b ounded Ω

⊂ Ω.

Proof. The proof is an easy consequence of Theorem IV.4.1, and Theorem

IV.5. 1, if one bears i n mind that (VII.1.2) can be viewed as a weak form of

the Stokes equatio n with f replaced by R(f −

∂v

∂x

). ut

In the remaining part of this section we shall be concerned with the unique-

ness of generalized solutions. Such a study is slightly more complicated than

the analo gous one for the Stokes problem. To see why, let v and w denote

two generalized solutions correspondi ng to the same data. Setting u = w −v,

from (VII.1.1) we obtain that u obeys the identity

F(ϕ) ≡ (∇u, ∇ϕ) − R(

∂u

∂x

, ϕ) = 0, f or all ϕ ∈ D(Ω). (VII.1.3)

Assuming Ω locally Lipschitz, as in the case of Stokes problem, we easily show

that u ∈ D

1,2

(Ω). However, it is not obvious that we can replace in (VII.1. 3),

ϕ with u, nor is it obvious that, even if this replacement is permitted, we can

conclude that

(

∂u

∂x

, u) = 0 . (VII.1.4)

We observe that these assumptions are deﬁnitely satisﬁed by any q-weak solution.

Actually, they are implied by the following one:

v ∈ L

loc

(Ω) , ∇v ∈ L

loc

(Ω), with v satisfying ( VII.1.1) for all ϕ ∈ D(Ω).

In f act, under these conditions, by Lemma II.6.1, we have v ∈ W

1,q

loc

(Ω) and then,

by Lemma VII.1.1, it follows p ∈ L

loc

(Ω); see also Remark VII.1.2.

By Remark VII.1.2, this latter condition is certai nly satisﬁed by any q-weak

solution under the stated assumption on Ω.

VII.1 Generalized Solutions. Regularity and Uniqueness 423

Notwithstanding, i f f ∈ D

−1,2

(Ω), one has that the functional

ϕ ∈ D(Ω) → (

∂u

∂x

, ϕ) ∈ R

can b e extended to a (bounded) li near functional, δ

u, on D

1,2

(Ω); see Remark

VII.1.2 and Theorem II.1 .7. Since, clearly, (∇u, ·) deﬁnes a (bounded) linear

functional, A(u), on D

1,2

(Ω) one can then equivalently rewrite (VII.1.3) in

the following abstract fo rm

A(u) − Rδ

u = 0 in D

−1,2

(Ω). (VII.1.5)

Now, Gal di (2007, Proposition 1 .2) shows, in a diﬀerent context, the “ab-

stract” counterpart of (VII.1.4 ), namely, [δ

u, u] = 0. Once we employ this

information back in (VII.1.5), we immediately ﬁnd [A(u), u] ≡ |u|

1,2

= 0,

which, in turn, implies u(x) = 0, for all x ∈ Ω.

Here, in order to show uniqueness, we will use a diﬀerent argument, based

on the asymptotic behavior of solutions to (VI I.1.3), that will be completely

justiﬁed in Section VII.6. Another, still diﬀerent, approach will be presented

in Section VIII.2 for the more general case of g eneralized Oseen problem.

We begin to observe that, from Theorem VII.1.1 it follows that u and the

corresponding pressure ﬁeld π, say, are inﬁnitely diﬀerentiabl e in Ω so that

(VII.1.3 ) can be written pointwise:

∆u + R

∂u

∂x

= ∇π

∇ · u = 0.

(VII.1.6 )

Furthermore, for any R > δ(Ω

), from Theorem II.4.2 we ﬁnd the existence

of a sequence {u

} ⊂ C

∞

(Ω

) vanishing near ∂Ω fo r all k ∈ N and approx-

imating u in the norm of the space W

1,2

(Ω

). Multiplying (VII.1 .6) by u

and integrating by parts over Ω

we easily deduce

Ω



∇u : ∇u

−R

∂u

∂x

· u



∂B

n ·



∇u · u

− πu



where n is the outer normal to ∂B

. We now let k → ∞ into this relation,

and recalling that u, π ∈ C

∞

(Ω), with the aid of Theorem II.4.1 we deduce

|u|

1,2,Ω

−

Ω

∇ · (u

) =

∂B

n · {∇u ·u − πu}.

We next apply the results of Exercise I I.4.3 to the second integral on the left-

hand side of this identity and recall that u has zero trace at ∂Ω to recover

|u|

1,2,Ω

∂B

n ·



∇u · u +

−πu



. (VII. 1.7)

424 VII Steady Oseen Fl ow in Exterior Domains

In Theorem VII.6.2 of Section VII.6 it will be proved that every suﬃciently

smooth solution to the Oseen system corresponding to a body force of compact

support and having a certain degree of summability at inﬁnity must decay

there in a suitable way. In particular, such a theorem ensures for u and π the

following estimates for every large R (see Exercise VII.6.1)

∂B



∇u : ∇u + u



≤ cR

−(n−1)/2

∂B

≤ cR

−(n−1)

(VII.1.8 )

Then, employing the Schwarz inequality on the right-hand side of (VII.1.7),

using (VII.1 .8), and letting R → ∞ we conclude u ≡ 0.

We have thus proved

Theorem VII.1.2 Let Ω be locally Lipschitz and let v be a generalized solu-

tion to (VI I.0.2), (VII.0.3) corresponding to f ∈ W

−1,2

(Ω

), Ω

any bounded

subdomain with Ω

⊂ Ω, a nd v

∗

∈ W

1/2,2

(∂Ω). Then, if w is another gener-

alized solution corresponding to the same data , it is v ≡ w.

Remark VI I.1.3 Theorem VII.1.2 wil l be extended to the case of arbitrary

q-generalized solutions (q 6= 2) in Exercise VII.6.2. 

VII.2 Existence of Ge neraliz ed Sol utions for

Three-Dimensional Flow

This section is devoted to proving existence of generalized solutions when Ω

is a three-dimensional domain, the two-dimensional case being postponed to

Section VII.5; see also Remark VII.2.1. To reach this goal, we begin to ob-

serve that, unlike for the Stokes problem, we can no longer employ the Riesz

representation theorem, since the left-hand side of (VII.1.1) does not deﬁne a

symmetric form for all R 6= 0 . We shall then use ano ther method which, in-

terestingly enough, though introduced by B.G. Galerkin in 1915 for studying

linear problems, was used in the ﬂuid dynamical context directly in the nonlin-

ear case at the beginning of the ﬁfties and sixties by E. Hopf and by H. Fuji ta,

respectively, and only in 1965 was it used by R. Finn in linearized approxi-

mations of the Navier–Stokes equations. To apply this method, however, we

need a preliminary result concerning the existence of a special complete set

in D

1,2

(Ω).

Lemma VII.2.1 Let Ω be an arbitrary domain of R

, n ≥ 2. Then, there ex-

ists a denumerable set of functions {ϕ

} whose linear hull is dense in D

1,2

(Ω)

and has the following properties

(i) ϕ

∈ D(Ω), for all k ∈ N;

VII.2 Existence for Three-Dimensional Flow 425

(ii) (∇ϕ

, ∇ϕ

) = δ

or (ϕ

, ϕ

) = δ

, for all k, j ∈ N;

(iii) Given ϕ ∈ D(Ω), and κ ∈ N, for any ε > 0 there exist m = m(ε) ∈ N

and γ

, ..., γ

∈ R, such that

k∇ϕ −

i=1

∇ϕ

+ kρ(ϕ −

i=1

< ε ,

for all s ≥ 2 , where ρ = (|x| + 1)

κ/s

Proof. Let H

0,ρ

(Ω), with ` > n/2 + 1, be the completion of D(Ω) in the norm

kϕk

`,2,ρ

≡ kρϕk

+ kϕk

`,2

Clearly, H

0,ρ

(Ω) is a subspace o f W

`,2

(Ω). Moreover, i t is also isomorphic to

a closed subspace of [L

(Ω)]

, for suitable N = N (`, n), via the map

ϕ ∈ H

0,ρ

(Ω) →



ρ ϕ

, . . . , ρ ϕ

; (D

)

1≤|α|≤`

; . . . ;

)

1≤|α|≤`



∈ [L

(Ω)]

Thus, in particular, H

0,ρ

(Ω) is separable (see Theorem II.1.5), and so i s its

subset D(Ω) (see Theorem II.1 .1). As a consequence, there exists a basis

in H

0,ρ

(Ω) of functions from D(Ω), which we will denote by {ψ

}. Since

0,ρ

(Ω) ,→ D

1,2

(Ω), the linear hull of {ψ

} must be dense in D

1,2

(Ω) as well.

Take ϕ ∈ D(Ω) and ﬁx ε > 0; there exist N = N (ε) ∈ N and α

, . . . , α

∈ R

such that

kϕ −

i=1

`,2,ρ

< ε.

By the embedding Theorem II.3.2, it follows that

kϕ −

i=1

< c ε

with c = c(Ω, n, `). We may orthonormalize {ψ

} in D

1,2

(Ω) or in L

(Ω)

by the Schmidt procedure, to obtain another denumerable set {ϕ

} whose

linear hull is still dense in D

1,2

(Ω). Since every ϕ

is a linear combination of

, . . . , ψ

and, conversely, every ψ

is a linear combination of ϕ

, . . . , ϕ

, it

is easy to check that the system {ϕ

} satisﬁes all the statements in the l emma

which is thus completely proved. ut

We are now in a position to prove the following.

Theorem VII.2.1 Let Ω be a three-dimensional exterior, locally Lipschitz

domain. Given

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2,2

(∂Ω) ,

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

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