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

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796 XI Three-Dimensional Flow in Exterior Domains. Rotational Case

this connection, we would like to recall also the contribution of Silvestre (2004,

Theorem 3.1). Unlike Corollary XI.1.1, valid for any generalized solution, this

latter author proves the existence of at least one generalized solution satisfying

the pointwise property (XI.1.16).

Section XI.2. All results in this section are due to me.

Section XI.3. A proof of existence of generalized solutions under the as-

sumption that the boundary datum v

∗

(is suﬃciently smooth and) has zero

total ﬂux through ∂Ω is due to Leray (1933, Chapter III). If v

∗

reduces to

a ri gid motion, namely, v

∗

= v

+ v

× x, v

, v

∈ R

, a simpler proof is

given in Borchers (1992, Korollar 4.1). However, a similar result and under

the same boundary conditions can be easily deduced also from earlier work of

Weinberger (19 73) and Serre (1987).

The existence theory i n its full generality, as presented in Theorem XI.3.1,

is due to me.

Section XI.4. The main result, Theorem XI.4.1, is taken from Gal di & Kyed

(2011a , Theorem 4.4). As already pointed out in Remark XI.4.3, we would like

to emphasize one more time the interesting op en question whether v + v

∞

is square-summable in a neighborhood of inﬁnity (that is, whether the liquid

possesses a ﬁnite kinetic energy), in the case that v

∗

≡ f ≡ 0 and v

·ω 6= 0.

Section XI.5. All results in this section are due to me. Simil ar results, under

more stringent assumptions on the data, can be found in Galdi & Kyed (2010,

2011a).

Section XI.6. The main results presented in Theorem XI.6.1–Theorem XI.6.3

are due to Galdi & Kyed (2011a). However, the proof of Theorem XI.6 .1 given

here diﬀers in som e signiﬁcant details from the analogous one furnished by

the above authors.

The signiﬁcant problem that is left open is the determination of leading

terms (if any) in the asym pto tic expansion of the velocity and pressure ﬁelds.

Another interesting problem that deserves attention is the asymptotic be-

havior of the vorticity ﬁeld. It is very likely that outside the “wake region”

the vorticity decays exponentially fast. Nevertheless, a proof of this property

is far from being obvious.

Section XI.7. Theorem XI.7 .1 is basically due to Galdi (2003) and Galdi &

Kyed (2010).

The fa ct that a suitable Landa u solution is the leading term in the asymp-

totic expansion of the velocity ﬁeld was ﬁrst discovered by Farwig & Hishida

(2009). In fact, the mai n result of these authors, under assumptions slightly

diﬀerent from those of Theorem XI.7.4, consists in the proof o f a representa-

tion for v similar to (XI.7.9), where, however, the quantity v + v

∞

− U

estimated in Lebesgue spaces rather than pointwise.

XII

Steady Navier–Stokes Flow in

Two-Dimensional Exterior Domains

F.F. CHOPIN, Scherzo op.31, bars 1-3.

Introduction

In this chapter we shall study plane steady ﬂow occurring in the complement

of a compact region. Speciﬁcally, we shall investigate existence, uniqueness

and asymptotic behavior of solutions v, p to the Navier–Stokes system:

∆v = Rv · ∇v + ∇p + Rf

∇· v = 0

)

in Ω

v = v

∗

at ∂Ω

(XII.0.1 )

where Ω is an exterior domain of R

. As in the previous chapter, (XII.0.1) is in

a nondimensional form, with R representing the Reynolds number. To system

(XII.0.1 ) we must append the conditi on at inﬁnity on the velocity ﬁeld, tha t

we choose to be of the form

As we mentioned previously in several occasions, the steady-state, two-

dimensional exterior problem in a rotating frame still lacks of signiﬁcant results.

For this reason, it will not be treated in this monograph.

797G.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_12, © Springer Science+Business Media, LLC 2011

798 XII Two-Dimensional Flow in Exterior Domains

lim

|x|→∞

v(x) = −v

∞

, (XII.0.2 )

with v

∞

a prescribed (constant) vector of R

. As already pointed out, a

signiﬁcant application of problem (XII.0.1)–(XII.0.2) occurs when f ≡ v

∗

≡ 0

and v

∞

6= 0, and regards the steady-state motion of a viscous liquid around a

long, straight cylinder C with axis a, assuming that the liquid is at rest at large

distances from C, and that C moves with (constant) translational velocity v

∞

orthogonal to a. Actually, in a region of ﬂow suﬃciently far from the two ends

of C and including C, one may expect that the velocity ﬁeld of the liquid is

independent of the coordinate parallel to a and, moreover, that there is no

motion in the direction o f a. Therefore, the relevant region of ﬂow can be

reasonably approximated by a two-dimensional domain placed in a suitable

plane orthogonal to a and exterior to the cross-section of C.

Using the methods of Section X.4 (cf. Remark X.4.4), we show that if Ω is

locally Lipschitz with ∂Ω 6= ∅, and if the ﬂux of v

∗

through ∂Ω is suﬃciently

“small” in the sense of (X.4.47), then (XII.0.1) admits at least one sol ution

v, p, without further restrictions on the size of the data. Such a solution is

also smooth if Ω and the data are likewise smooth. Concerning the behavior

at large distances, the only informa tion one can obtain is that v has a ﬁnite

Dirichlet integral, that i s,

Ω

∇v : ∇v ≤ M, (XII.0.3)

with M = M (Ω, f, v

∗

, R). However, (XII.0.3) alone is not enough to ensure

that v tends (even in a generalized sense) to a constant vector to inﬁnity, as

is shown by simple counterexamples; see Section XII.1.

Therefore, we don’t

know if these solutions verify condition (XII.0.2). Furthermore, if v

∗

≡ f ≡ 0,

we do not know, in general, if this solution is nontrivial.

A separate consid-

eration deserves the case Ω = R

. Actually, in such a case, even the existence

of solutions to the system (XII.0.1) is not known, in general.

Technically, this

is due to the fact that a sequence of functions obeying the bound (XII.0.3)

with Ω = R

, need not be convergent in any of the spaces L

), q > 1,

R > 0 (cf. Exercise II.7.3), and this in turn implies that one can not show

the convergence of the nonlinear term along the sequence of approximating

solutions; see Remark X.4.4 .

The investigatio n of whether solutions to (XII.0.1 ) satisfying (XII.0.3) may

also obey (XII.0.2) has a ttracted the attention of many writers; see Section

XII.3. Speciﬁcally, Gilbarg & Weinberger (1974, 1978) have shown that if f is

A sharp estimate at large distances of functions satisfying condition (XII.0.3) has

been given in Theorem II.9.1.

If Ω

is symmetric around some direction, then one can construct nontrivial,

suitably symmetric solutions; see the Notes for this Chapter.

Existence can be established if the data satisfy suitable symmetry requirements;

see the Notes for this Chapter.

XII Introduction 799

of bounded support, every v satisfying (XII.0.1) for a suitable pressure p, and

(XII.0.3 ), either converges at large distances, in a well-deﬁned sense, to some

constant vector v

, or that the L

-norm of v over the unit circle approaches

inﬁnity at large distances. However, they can not show that v

= −v

∞

. If

∗

≡ f ≡ 0, Amick (1988) has proved that v ∈ L

∞

(Ω). Nevertheless, also i n

such a case, one cannot infer that v

= −v

∞

if v

∞

6= 0 and, wha t perhaps

is more surprising, if v

= 0, one is able to conclude that v ≡ 0 only in the

special situation where Ω = R

At this point we may wonder if we could prove existence by means of

diﬀerent techniques, such as a ﬁxed po int argument. This problem, which was

ﬁrst considered by Finn & Smith (196 7b), is in fact solvable when the velocity

ﬁeld v

∞

is not zero and the data are suﬃciently sm all; see Section XII.5. It

is interesting to observe that the corresponding solutions obey, in particular,

condition (XII.0.3). It should also be empha sized that if v

∞

= 0, the above

techniques do not work and so it is no t known whether (XII.0.1), (XII.0.2)

with v

∞

= 0 is resoluble, even for small data.

The reason for this unequal

result i s essentially due to the fact that the approach followed is based on

ﬁxed point arguments that rely on linearized versions of (XII.0.1), (XII.0.2).

As we know from Chapters V and VII, the linearization when v

∞

6= 0 (Oseen

system) produces solutions that, in the neighborho od of inﬁnity, are more

regular than those corresponding to the linearization when v

∞

= 0 (Stokes

system). In this respect, we recall that a simila r circumstance occurs also for

three-dimensional ﬂows.

In view of all the above, it is natural to ask whether (XII.0.1)–(XII.0 .2)

may indeed admit a solution for data of arbitrary si ze. This question has

been investigated by Galdi (1999b) for the physically relevant problem where

∗

≡ f ≡ 0, which, as we noted previously, describes the motion of the liquid

past a suﬃciently long cylinder. In such a case the data are represented by

the translational velocity of the cylinder that, in dimensionless form, is given

by Re

(assuming v

∞

). Let us call P this parti cular problem. “Arbitrary

data” for P means then “arbitrary speed” of the cylinder, namely, any R > 0.

Galdi has shown tha t if Ω possesses an axis of symmetry (e.g., Ω is the exterior

of a circle) then, if there exists R > 0 such that for all R ≥ R problem P

has no solution in a very general regularity class, the homogeneous problem

obtained by setting in (XII.0.1)–(XII.0.2) v

∗

≡ f ≡ v

∞

≡ 0, and R = 1

must admit a non-zero solution, a fact that is very questionable on physical

grounds; see Section XII.6.

Another question that arises is the asymptotic structure of a ﬁeld v satis-

fying (XII.0.1)–(XII.0.3), for suitable p. One may expect that v can be rep-

resented asymptotically by an expansion in “reasonable” functions of r ≡ | x|

with coeﬃcients independent of r. However, if v

∞

= 0, not every such solution

can be represented in this way, for one can exhibit examples, due to Hamel,

of solutions to (XII.0.1), (XII. 0.2) with v

∞

= 0 that obey (XII.0.3) and decay

See, however, the Notes for this Chapter for some partial results.

800 XII Two-Dimensional Flow in Exterior Domains

more slowly than any negative power of r; see Section XII.6. If v

∞

6= 0 , how-

ever, we have a result completely analogous to the three-dimensional case,

namely, if v satisﬁes (XII. 0.1)–(XII.0.3), for a suitable pressure p, and, in

addition, f is (suﬃciently smooth and) of bounded support, then v has the

same asymptotic structure of the Oseen fundamental solution. The proof of

this result, based on the papers of Galdi & Sohr (1995) and Sazonov (1999),

is due to Sazonov; see Section XII.6 and Section XII.8.

In the last section of the chapter we shall study the behavior of solutions

to (XII.0.1), (XII.0.2) in the limit of vanishing Reynolds number for the case

f = 0.

This problem is considered in the class C of solutions whose existence

has been determined in Section XII.5. We show that the solutions from C

tend (in an appropriate sense) to solutions of the Stokes probl em obtai ned

by formally taking R = 0 into (XII.0.1), (XII. 0.2). The interesting feature of

this study is that, as expected from the linear theory (cf. Section VII.8), the

limit process is singular in that it does not preserve the condition at inﬁnity

(XII.0.2 ). In fact, following the work of Ga ldi (1993), we show that the limit

solution satisﬁes (XII.0.2) if and only if v

∗

veriﬁes a suitable condition. In

the case when Ω is the exterior of a unit circle, this condi tion reads:

∂Ω

∗

+ v

∞

) = 0.

Notation. Throughout this chapter Cartesian coordinates of a point x, are

denoted either (as usual) by x

, x

, or (more simply) by x,y. When using the

latter notation, the components of a vector w are written as w

, w

. We shall

also frequently employ polar coordinates, r,θ. In such a case, the components

of w, are denoted, as customary, by w

, w

XII.1 Gene ralized Solutions and D-Soluti ons

As already observed in Remark X.4.4, if Ω is a l ocally Lipschitz exterior do-

main in the plane with ∂Ω 6= ∅, and if the ﬂux of v

∗

through ∂Ω satisﬁes

(X.4.47), we may use the same method employed in Section X.4 to construct,

for all R > 0, a vector ﬁeld satisfying conditions (i)-(iii) and (v) of Deﬁnition

IX.1. 1, together with the energy inequali ty (IX.4.16). Moreover, we can asso-

ciate to v a pressure ﬁeld p (cf. Lemma X.1.1) and both v and p are smooth

provided Ω and the data are smooth; cf. Theorem X.1.1. However, v and p

solve in principle only problem (XII.0.1) because, unlike the three-dimensional

case, the property (i), namely,

Ω

∇v : ∇v < ∞, (XII.1.1)

is not enough a priori to control the behavior at inﬁnity of v. Thus, the limiting

condition (XII.0.2) or even the weaker condition (iv) of Deﬁnitio n X.1.1 need

This assumption is made for the sake of simplicity.

XII.2 On the Uniqueness of Generalized Solutions 801

not be satisﬁed. That the problem of proving the convergence at inﬁnity of a

solution v under the sole information (XII.1.1) is far from being obvious, i s

shown by the fact that there are ﬁelds w in Ω having a ﬁnite Dirichlet integral

and that become unbounded at large distances like a power of log |x|. Take,

for instance, Ω = R

− B(0) and w(x) = (log |x|)

, 0 < α < 1/2. Therefore,

in order to prove the convergence of a solution v at inﬁnity, the equations of

motion must play a f undamental role.

The situation becomes completely obscure in the case Ω = R

, where the

methods of Section X.4 do not even furni sh existence of a vector ﬁeld satisfying

(XII.0.1 ) for a suitable pressure p; see Remark X.4.4.

In what follows, we will make a deﬁnite distinction between a generalized

solution v to (XII.0.1), (XII.0.2) in the sense of Deﬁnition X.1.1, and a ﬁeld

v sati sfy ing only (i)-(iii) and (v) of the same deﬁnition. In this latter case,

the vector ﬁeld v will be referred to as a D-solution, in spite of the fact that

the word “solution” is not the most appropriate, since v need not verify the

condition at inﬁnity. A study on the asymptotic behavior of D-solutions i s

performed in Section XII.3, where it will be shown, among other things, that,

as |x| → ∞, v(x) tends in a suitable sense to some vector v

. Whether this

vector coincides with the vector −v

∞

prescribed in (iv) of Deﬁnition X.1.1

remains, however, a fundamental open question.

Nevertheless, we may wonder if, by using a diﬀerent technique, one can

show existence, at least in the range of small data. As shown in Section XII.5,

this is indeed possible, provided v

∞

is not zero. If v

∞

is zero, the questi on of

existence of generalized solutions, even with small data, is open and, probably,

for “generic” data, it does not admi t a positive answer; see also the Notes for

this Chapter.

XII.2 On the Uniqueness of Generalized Solutions

Maybe uniqueness of generalized solutions is a more complicated question

than existence itself. I ndeed it represents a formidable problem that, for its

resolution, requires in my opi nion the contribution o f completely new ideas

and methods. To test the diﬃculty, let v

, p

and v

, p

be two generalized

solutions to problem (XII.0.1), (XII.0.2), which will be assumed smooth for

simplicity. Letting

u = v

− v

π = (p

−p

)/R

we have

∆u = u · ∇u + v

· ∇u + u · ∇v

+ ∇π

∇ · u = 0







in Ω

u = 0 at ∂Ω

lim

|x|→∞

u(x) = 0.

(XII.2.1 )

802 XII Two-Dimensional Flow in Exterior Domains

Multiply (XII.2.1)

by u, integrate by parts over Ω

and use (XII.2.1)

obtain

−1

Ω

∇u : ∇u = −

Ω

u·∇v

·u+

∂B

[u·

∂u

∂n

−

(u+v

)·n−πu ·n].

(XII.2.2 )

Now assume tha t u, v

, and π behave in such a way that the surface integral

tends to zero as R → ∞. (This is not known if v

∞

= 0, while, if v

∞

6= 0,

it is true for any generalized solution which satisﬁes some further condition;

cf. Section XII.8.) From (XII.2.2) in the limit R → ∞ we ﬁnd

−1

Ω

∇u : ∇u = −

Ω

u · ∇v

· u. (XII. 2.3)

This is the classical relation, which we have used several times to show unique-

ness o f generalized solutions, sometimes with the equality sign replaced by the

inequality one. However, for the case at hand, relation (XII.2.3) is not g oing

to produce uniqueness for smal l R. Actually, we would need a n estimate of

the type

−

Ω

u · ∇v

· u ≤ c

Ω

∇u : ∇u (XII.2.4)

for som e c = c(v

, Ω) ∈ (0, ∞), to conclude

Ω

∇u : ∇u = 0, for R < c

−1

and hence uniqueness. However, the validity of (XII.2.4) is very unlikely,

no matter what summabil ity assumptio ns on u and v

are made and, con-

sequently, the “traditional” method gives no information whatsoever about

uniqueness, even at small Reynolds numbers.

Nevertheless, what is certainly true i s that, if R is “ suﬃciently” large and

the ﬂux through the wall ∂Ω is not zero, uniqueness of generalized solutions is

lost, as can be seen by means of sim ple examples. To show this, let us consider

an indeﬁnite, circular cylinder C of radius r

immersed in a viscous liquid and

assume that the radial comp onent of the velocity of the liquid, when evaluated

at the lateral surface of C, is a constant V. The appropriate Reynolds number

is then deﬁned by

R = V r

/ν.

Using V and r

as reference velocity and length, respectively, from (XII.0.1),

(XII.0.2 ) we obtain that the two-dim ensional moti ons of the liquid in every

plane orthogonal to C, w hen referred to polar coo rdinates (r, θ), must obey

the following problem

From (I I.5.7) it follows that (XII.2.4) holds if ∇v

(x) = O(|x|

−2

log

−2

|x|).

However, even admitting that v

at large distances has the same behavior

as the solution of the corresponding linearized problem, we would have only

∇v

(x) = O(|x|

−1

); cf. Section V.3 and Section VII.6.

XII.2 On the Uniqueness of Generalized Solutions 803



∂v

∂r

∂v

∂θ

−



= −

∂p

∂r

+ ∆v

−

∂v

∂θ

−



∂v

∂r

∂v

∂θ



= −

∂p

∂θ

+ ∆v

∂v

∂θ

−

∂v

∂r

∂v

∂θ

= 0











in Ω

v(1, θ) = v

∗

lim

r→∞

v(r, θ) = −v

∞

(XII.2.5 )

where v = (v

, v

), Ω is the exterior of the unit circle, v

∗

(θ) is prescribed and

∆ =

∂

∂r



∂

∂r



∂

∂θ

Now, by a simple computation, we show that for any choice of γ, ω ∈ R and

R > 1, R 6= 2, problem (XII.2.5) with

∞

= 0, v

∗

= (−R, γ − ω/(R− 2)) (XII.2.6)

admi ts the elementary solution

= −



γ −

R−2

−R+2



p = R



−



(XII.2.7 )

The ﬂow (XII.2.6), (XII.2.7) which was discovered by Hamel (1916, pp. 51-52)

has a simple physical explanation; cf. Preston (1950). Speciﬁcally, it represents

the motio n of a liquid subject to a uni t suction in the direction orthogonal to

the wall of the circular cylinder, while the cylinder rotates with the angular

velocity

γ −

R− 2

By the arbitrarity of R, γ, and ω, we may choose

ω = γ(R − 2) (XII.2.8)

and the solution (XII.2.7) special izes to

Notice that (XII.2.7) is not a solution of the corresponding linearized Stokes

problem, that is, (XII.2.5) with R = 0, since (XII.2.5)

is violated.

804 XII Two-Dimensional Flow in Exterior Domains

= −

R− 2



1 − r

−R+2



p = R



−



(XII.2.9 )

Moreover, the data are independent of γ, since with the choice (XII.2 .8) and

for R > 1 and R 6= 2, conditions (XII.2.6) become

∞

= 0, v

∗

= (−R, 0). (XII.2 .10)

As a consequence, we conclude that if R > 1 and R 6= 2 the ﬁelds (XII.2.9)

constitute a one-parameter family of solutions to (XII.2.5), parameterized in

ω, and corresponding to the same da ta (XII.2.10). Furthermore, since the

gradients of the velocity ﬁelds square summable in Ω, they are generalized

solutions.

The example just given suggests that, in the exterior two-dimensional

problem, the prescription of the velocity ﬁeld alone at the boundary and at

inﬁnity is not in general enough to secure the uniqueness o f the solutio n, and

other extra conditions may be needed. For instance, in the class (XII.2.9)–

(XII.2.1 0) uniqueness is recovered if we prescribe at the boundary r = 1 the

vorticity ω = r

−1

∂(rv

)/∂r.

XII.3 On the Asymptotic Behavior of D-Solutions

The aim of this section is to investigate the behavior at large distances

of D-solutions, namely, of vector ﬁelds v satisfying (XII.0.1) and condition

(XII.1.1 ).

Since we are interested onl y in the regularity at inﬁnity, we shall suppose

that the D-solutions with which we a re dealing are as smooth as required

by the formal manipulation we shall perform. Of course, to substantiate this

procedure, it is enough to assume that the data and the domain Ω have a

suﬃcient degree of regularity, what degree will b e tacitly understood. More-

over, since the results we shall ﬁnd are essentially independent of the Reynolds

number R, we shall put, for simplicity, R = 1.

We wish now to collect some notation and fo rmulas that will be frequently

used. T he scalar ﬁeld

ω =

∂w

∂y

−

∂w

∂x

is the vorticity of a vector ﬁeld w. From (XII.0 .1), it follows that the vorticity

of a D-solution v veriﬁes the equation

∆ω − v · ∇ω =

∂f

∂y

−

∂f

∂x

. (XII.3.1)

XII.3 On the Asymptotic Behavior of D-Solutions 805

Owing to the solenoidality of v, we also have

∆v

∂ω

∂y

, ∆v

= −

∂ω

∂x

. (XII.3.2)

The fo llowing result is based on ideas of Gilbarg & Weinberger (1978, Lemma

2.3).

Lemma XII.3.1 Let v be a D-solution to (XII.0.1) with f ∈ L

(Ω

), some

ρ > δ(Ω

). Then, ∇ω ∈ L

(Ω

) and the following identity holds

Ω

|∇ω|

∂B



∂ω

∂n

− v · nω



Ω



∂ω

∂x

−f

∂ω

∂y



Proof. Let h ∈ C

(R) be a positive function with piecewise diﬀerentiable

ﬁrst derivatives and let ψ

be the Sobolev “cut-oﬀ” function (II.6.1 ) with

exp

√

ln R > ρ. Using (XII.3.1), we show the following identity

∇ · [ψ

∇h − h∇ψ

−ψ

hv] = ψ

|∇ω|

− h (∆ψ

+ v · ∇ψ

)

+ h



∂f

∂y

−

∂f

∂x



(XII.3.3 )

where prime means diﬀerentiation. Setting

∗

= max

x∈∂B

|ω(x)|,

we choose

h =

(

if |ω| ≤ ω

(2|ω| −ω

) if |ω| ≥ ω

(XII.3.4 )

for ω

≥ ω

∗

. From (XII.3.3) and (XII.3.4) we easily recover

Ω

|∇ω|

Ω

h(∆ψ

+ v · ∇ψ

) +

∂B



∂ω

∂n

−v ·nω



Ω





∂h

∂x



− f



∂h

∂x



(XII.3.5 )

Taking into account that

h ≤ min



, 2 ω

|ω|



and the properties (II.6.2), (II.6.5), and (II.6.6

) of the function ψ

, we ﬁnd