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

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646 IX Steady Navier–Stokes Flow in Bounded Domains

Ω

w · ∇w · V = 1. (I)

Note that if V is an extension of v

∗

, V + ϕ is such for all ϕ ∈ D(Ω), and

from (I) it follows that

Ω

w · ∇w · ϕ = 0 for all ϕ ∈ D(Ω).

In view of Lemma III.1.1, this latter condition, together with the fact that w is

solenoidal and vanishes at the bounda ry, tells us that w solves the Euler-type

problem:

w · ∇w = ∇π

∇ · w = 0

)

in Ω

w = 0 at ∂Ω

(II)

for some “pressure ﬁeld” π. Therefore, the uniform bound (∗∗), and hence

existence of solutions to (IX.0.1), (IX.0.2), wi ll be proved whenever we show

that conditions (I) and (II) are incompatible. It is readily seen that (I) and

(II) are certainly incompatible if the stronger condi tion (IX.4.7) is satisﬁed.

Actually, from (II)

1,3

it follows that π is a constant π

(say) on each connected

component Γ

of ∂Ω, i = 1, . . . , m+1.

Multiplying (II)

by V and integrating

by parts over Ω we ﬁnd

Ω

w · ∇w · V =

m+1

i=1

∗

· n

and so, if we assume (IX.4.7), we obta in

Ω

w · ∇w · V = 0,

which contradicts (I). Incompatibility under the sole condition (IX.4.6) was

left open by Leray. However, it is readily seen that Leray’s approach as it

stands does not lead in general to a contradiction. In f act, there a re examples

where conditions (I) and (II) are not incompatible. Consider, for instance, the

annular domain

Ω =



x ∈ R

: R

< |x| < R



(III)

and set

Φ =

∗

· n = −

∗

·n,

where



x ∈ R

: |x| = R



, Γ



x ∈ R

: |x| = R



A detailed proof of this property is given by Kapitanski

i & Pileckas (1983).

IX.7 Notes for the Chapter 647

It is readily seen that the ﬁeld u = u(r)e

solves (II) with π = −

/r).

Moreover, taking into account (IX.4.9 ), we show

Ω

u · ∇u · V = −Φ

As a consequence, if Φ < 0 the ﬁeld

w =



−Φ



1/2

satisﬁes both conditions (I) and (II) that are, therefore, compatible.

The contradiction method of Leray just described has been more recently

used by Kapitanski

i & Pileckas (1983, §4), Borchers & Pileckas (1994) and, i n-

dependently, by Amick (1984). In particular, the latter author pushes Leray’s

argument a bit further to show that conditions (I) and (II) are indeed in-

compatible in a class of plane domains Ω a nd ﬁelds v

∗

, w possessing certain

suitable symmetry. As a result, under these assumptions on the data, Am-

ick proves existence requiring o nly the compatibility condition (IX.4.6) on

the ﬂux. The result of Amick has been successively rediscovered by Sazonov

(1993). Another, constructive proof of Amick’s result can be found in Fu-

jita (1998). If Ω is the annulus (III), Morimoto (1992) and Morimoto & Ukai

(1996) have furnished existence under the general condition (IX.4.6 ), provided,

however, the boundary data satisfy some extra restrictions. In this respect,

see also Morimoto (1995), Fujita & Morimoto (1997), and Russo & Starita

(2008).

All mentioned papers need a suﬃciently high degree of smoothness on

both v

∗

and Ω, e.g., of class C

. The ﬁrst general result with v

∗

in the

“natural” trace space W

1/2,2

(∂Ω) is due to Foia¸s & Tema m (1978), under the

assumption that Ω is C

-smooth.

Existence results in locally Lipschitz domai ns, and with boundary data of

lesser regulari ty than the W

1/2,2

(∂Ω) one, have been proved in the interesting

paper of Russo (2003). The approach followed by this author is based on

potential-theoretic methods which, in turn, rely upon previous work of Fabes,

Kenig & Verchota (1988).

A maximum modulus result for (bounded) locally Lipschitz three-dimensional

domains has been shown by Russo (2011).

Uniqueness under conditions of the type given in Theorem IX.4.2 but with

a computable constant c

can b e found i n Payne (1965).

Section IX.5. Regularity of generalized solutions was ﬁrst proved by La-

dyzhenskaya (1959b, Chapter II, §2). An independent proof is given by Fujita

(1961, §§4,5); see also Shapiro (1976a). Fujita also observes that analyticity

of a generalized solution (v, p), corresponding to an a nalytic f in Ω, follows

from the fact that v, p ∈ C

∞

(Ω) along wi th classical results of Morrey (1958 )

and Friedman (1958) on nonlinear analytic elliptic systems. A completely

648 IX Steady Navier–Stokes Flow in Bounded Domains

the Stokes problem, is due to Temam (1977, Chapter II, Propo sition 1.1).

All the above results hold in dimension n = 2, 3 but fail if n ≥ 4. The

problem of regularity in higher dimensions has been considered by von Wahl

(1978, Satz II.1) who gave a ﬁrst, partial answer for n = 4. Successively,

Gerhardt (1979) proved regularity of generalized solutions in dimension n = 4.

An analogous theorem has been later obtained, by diﬀerent tool s, by Giaquinta

& Modica (1982). Results found by all the above a uthors, however, do not

admi t a direct generalization to higher dimensions. Regularity of generalized

solution v in arbitrary dimension n ≥ 2 was ﬁrst proved by von Wahl (19 86)

as a by-product of his study on the unsteady Navier–Stokes equations; cf. also

Sohr & von Wahl (1984). von Wahl’s assumptions on v are a particular case

of those of Theorem IX.5.2, obtained by setting there s = 2. He also requires

extra regularity on Ω.

As a lready emphasized, our proof of smoothness of generalized (a nd q-

generalized) solutions is based on Lemma IX.5.1. Thi s lemma improves an

analogous result shown in Galdi (1994b, Chapter VIII, Lemma 5.1)

An interior regularity result simi lar to that of Theorem IX.5.1(a) is given

by Kim & Kozono (2006, Corollary 5). However, these authors require n ≥ 3

and p ∈ L

loc

(Ω), which are not needed in Theorem IX.5 .1(a).

In the paper of Kim & Kozono, the interesting problem of removable sin-

gularities is also addressed. More precisely, assuming that a solution (v, p)

to the steady-state Navier–Stokes equation is regular i n the punctured ba ll

B −{x

}, the q uestion is to ﬁnd conditions on the behavior of v(x) as x → x

that ensure that the solution is regular in the whole of B. Similar questions

were previously addressed by Dyer & Edmunds (1970), who ﬁrst studied the

problem, Shapiro (1974, 1976b, 1976c), and Choe & Ki m (2000).

The i mportant question of regularity of very weak solutions has been inves-

tigated by several authors. For interior regularity, we recall the contribution

of Kim & Kozono previously m entioned and its improved version furnished

by me in Theorem IX.5.1(a). Results on regularity up to the boundary can

be found in Maruˇs´ıc-Paloka (2000, Remark 3), Farwig, Galdi & Sohr (2006,

Corollary 1.5), Farwig & Sohr (200 9, Theorem 1.5), and Kim (2009, The-

orem 3). In particular, in the latter two papers, the authors independently

prove a result analo gous to Theorem IX.5.2(a), under the foll owing assump-

tions (i) v ∈ L

(Ω), q

= n, if n ≥ 3, q

> 2, if n = 2; (ii) ∇ · v = g,

g ∈ L

n+q

(Ω) ∩W

1,q

(Ω) with a suﬃciently “small” norm; (iii) v satisﬁes (∗∗)

for all ψ ∈ C

(Ω), and Ω of class C

2,1

(Farwig & Sohr), all ψ ∈

(Ω), and Ω

of class C

(Kim). All the above results leave open the intriguing question of

weather, for n = 2, a very weak solution, wi th v ∈ L

(Ω) and corresponding

to regular data, is regular.

different (and formally simpler) proof of regularity, based on the estimates of

Steady Navier–Stokes Flow in

Three-Dimensional Exterior Domai ns.

Irrotational Ca se

F.F. CHOPIN, Ballade op.23, bars 7-8.

Introduction

Objective of this and the next two chapters is to investigate the m athematical

properties of steady ﬂow of a v iscous incompressible ﬂuid that ﬁlls the entire

space outside a ﬁni te number of “bodies”, Ω

, . . . , Ω

, and whose motion is

governed by the fully nonlinear Navier–Stokes equations. More speciﬁcally, we

shall b e concerned with the following boundary value problem

ν∆v = v · ∇v + 2ω × v + ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω

(X.0.1)

to which we append the condition at inﬁnity

lim

|x|→∞

(v(x) + v

∞

(x)) = 0 . (X.0.2)

Here, as usual, Ω is a domain of R

, n = 2, 3, exterior to Ω

:= ∪

i=1

Ω

with Ω

compact, i = 1, . . . , s, and Ω

∩ Ω

= ∅, for i 6= j, representing the

649

G.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_10, © Springer Science+Business Media, LLC 2011

650 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

region of ﬂow, f and v

∗

are prescribed ﬁelds i n Ω and ∂Ω, respectively, while

∞

(x) := v

+ ω × x, with v

and ω given constant vectors. As explained in

Section I.2, this problem is of particularly great relevance in the study of the

steady ﬂow past a rigid body B that translates and rotates, when described

from a frame, S, attached to B. In such a case, s = 1, B ≡ Ω

, and ω

represents the (constant) a ngular velocity of the body, while v

is the velocity

of its center of mass referred to S, supp osed to be constant.

Since the type of diﬃculties encountered, the methods used and the results

found in solving problem (X.0.1)–(X.0.2) are diﬀerent according to whether

ω = 0 or ω 6= 0, and n = 2 or n = 3, we wish to treat the various cases in

as m any chapters. In the present one we shall consider the irrotational case

ω = 0, so that problem (X.0.1) reduces to

ν∆v = v ·∇v + ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω

(X.0.3)

along with the condition at inﬁnity

lim

|x|→∞

v(x) = −v

∞

(X.0.4)

where v

∞

≡ v

∈ R

With few exceptions, which will be either remarked on or explicitly noticed

in the statements of our results, we shall be concerned with three–dimensional

ﬂows, while we shall treat the analogous, more involved two-dimensional prob-

lem in Chapter XII. The three-dimensional case with ω 6= 0 will be studied

in Chapter XI. To date, no signiﬁcant results are available in two dimensions

when ω 6= 0.

We shall next describe the problems and the main results presented in

this chapter. Following the work of Leray (1933), Ladyzhenskaya (1959b), and

Fujita (1961), one gives, as we already did in previous chapters, a variational

formulation of the problem and proves the existence of a weak solution v to

(X.0.3), (X.0.4) without restrictions on the “size” of the data and even with a

nonzero (but small) total ﬂux of v

∗

through ∂Ω. Such solutions, characterized

by having a ﬁnite Dirichlet integral:

Ω

∇v : ∇v ≤ M, (X.0.5)

where M depends only on the data, are sometimes referred to as D-solutions,

or also Leray solutions. In the investigation of the features of D-solutions one

has to face two diﬀerent kinds of problems. On the one hand, similarly to

generalized solutions in bo unded domains, one has to analyze their diﬀeren-

tiability properties. However, this is simply achieved by using the regularity

See footnote 14 in Chapter I.

X Introduction 651

theory developed in the case Ω bounded and we can prove that D-sol utions

are, in fact, smooth provided the data are smooth. On the other hand, one

has to show that D-solutions have those properties expected from the physi-

cal point of view and tightly related to their behavior at large distances. For

example, they have to verify the energy equation:

2ν

Ω

D(v) : D(v) −

∂Ω

[ (v

∗

+ v

∞

) · T (v, p)

−

∗

+ v

∞

)

∗

] ·n +

Ω

f · (v + v

∞

) = 0

(X.0.6)

with D and T stretching and stress tensors (IV.8.6), which describes the

balance between the power of the work of external force, the work done on

the “body” Ω

, and the energy dissipated by the viscosity. Also, if f, v

∗

, and

∞

are “suﬃciently small” with respect to the viscosity ν, the corresponding

D-solutions must be unique. In addition, in the case s = 1, v

∗

= f = 0 (rigid,

impermeable body translating in the liquid with constant velocity v

), the

ﬂow must exhibit an inﬁnite wake extending in the direction opposite to v

∞

inside the wake the ﬂow is essentially vortical a nd the order of convergence of

v to v

∞

is diﬀerent depending on whether it is calculated inside o r outside

the wake. Finally, according to the boundary-l ayer concept, the ﬂow must be

potential outside the close vicinity of the body Ω

and of the wake, which

means (at least) that the vorticity should decay exponentially fast at large

distances and o utside the wake.

Despite of the eﬀorts of many mathematicians, for quite a long time these

questions had no answer to the point that, in 1959, R. Finn was led to in-

troduce another class of solutions characterized by the requirement that the

velocity ﬁeld v obeys, as |x| → ∞,

v(x) + v

∞

= O(|x|

−1/2−ε

), some ε > 0, if v

∞

6= 0

v(x) = O(|x|

−1

), if v

∞

= 0.

(X.0.7)

In a series of fundamental papers, Finn and his coworkers were then able to

show tha t any such solution has all the basic properties previously mentioned.

For this reason, he called solutions satisfying (X.0.7) physically reasonable

(P R). Moreover, in 1965, by extremely careful and painstaking estimates of

the Green’s tensor function, Finn showed that if the data are “small enough”

there exists a unique corresponding P R-solution.

It then appeared quite natural to study the relati on between a D-solution

and a P R-solution and, therefore, to ascertain if D-solutions can eﬀectively

describe the real world or are just mathematical inventions. However, although

it is a relatively simple task to prove that a P R-sol ution is a D-solution,

the question of whether the converse implication holds true has remained

open for years and, even today, it presents some aspects that are yet to be

clariﬁed. Speciﬁcally, if v

∞

6= 0, K.I. Bab enko (1973) has shown that every

652 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

D-solution v corresponding to a body force of bounded support is a P R-

solution (cf. also Galdi 1992b, Farwig & Sohr 1998) while Galdi (1992c) has

proved the same property for v

∞

= 0 provided, however, that v obeys a the

“energy inequality” (that is, (X.0.6) with “=” replaced by “≤”) and that

the viscosi ty is suﬃciently large. In this respect, it is worth noticing that

the class of D-solutions obeying the energy inequality is certainly nonempty.

The nonhomogeneity of these results for the cases v

∞

6= 0 and v

∞

= 0 is

essentially due to the following reason. They both rely on the asymptotic

properties of solutions to the linearized approximations of (X.0.3), (X.0.4).

Now, if v

∞

6= 0, such an a pproximation leads to the Oseen system, while if

∞

= 0 it leads to the Stokes system and we know from Chapters V and VII

that the asymptotic properties of solutions to the Stokes system are, in an

appropriate sense, weaker than those of the Oseen one.

In the present chapter, we shall follow the basic ideas of Galdi (1992a,

1992b, 1992c), with several changes in the details. Thus, after proving exis-

tence of D-solutions, we shall investigate, among other things, their a symp-

totic behavior at large distances and shall show that it coincides with that of

the Stokes or the Oseen fundamental tensor according to whether v

∞

is or is

not zero. However, if v

∞

= 0, we are able to prove this only for large viscosity

and for D-solutions obeying the energy inequal ity.

It should also be emphasized that if v

∞

= 0, several fundamental questions

remain open. For example, it is not known if, for given f and v

∗

there are

solutions obeying the energy equation (X.0.6) without restricti on on t he size

of the data or, equiva lently, for all values of the viscosity. Moreover, it is not

known i f, when Ω = R

, the only D-solution corresponding to f = 0 is the

zero solution; see Remark X.9.4.

In what follows, we shall ﬁnd it convenient to put (X.0.3), (X.0.4) into a

dimensionl ess form, and so we need comparison length d and velocity V . If

∞

6= 0 a nd |Ω

| 6= ∅, we can take d = δ(Ω

), V = |v

∞

| and so, introducing

the Reynolds number

R =

V d

the system (X.0.3) becomes

∆v = Rv · ∇v + ∇p + Rf

∇· v = 0

)

in Ω

v = v

∗

at ∂Ω

(X.0.8)

where v , v

∗

, p, and f are now nondimensional quantities. In such a case

the nondimensional velocity v

∞

becomes a unit vector that, without loss, we

will take coincident with the unit vector e

. If either v

∞

= 0 o r Ω ≡ R

this

choice of d and V is no longer possible, even though we can still give meaning

to (X.0.8), which is what we shall do throughout the chapter.

We wish to observe that, whenever possible, we shall explicitly remark if

and in what form a certain result we ﬁnd can be generalized to dimension

X.1 Generalized Solutions and Related Properties 653

n ≥ 4. If no comment is made, it will be tacitly understood that either the

result does not admit of a straightforward generalization to higher dimension

or, even, that such a generalization constitutes an open question.

Finally, we remark that all material presented here concerns the “cla ssi-

cal” formulation of the exterior problem. However, there is a wide variety of

exterior probl ems which can not be included within this formulation and are

of great physical relevance like, for example, the steady fall of a body or the

steady motion of a self- propelled body in a viscous liquid. We shall not treat

these questions and refer the i nterested reader to the works of Weinberger

(1972, 1973, 1974), Serre (1887), Galdi (19 99a), and to the comprehensive

article of Galdi (2002) and to the references cited therein.

X.1 Generalized Solutions. Preliminary Considerations

and Regularity Properties

We begi n to give a generalized formulation of the Navier–Stokes problem

(X.0.8) , (X.0.4) and to investigate some basic properties of the corresponding

solutions, including their regularity. To this end, assuming at ﬁrst v, p, and f

are suﬃciently smooth, we multiply (X.0.8)

by ϕ ∈ D(Ω) and integrate by

parts over Ω to obtain

Ω

∇v : ∇ϕ + R

Ω

v · ∇v · ϕ = −R

Ω

f · ϕ. (X.1 .1)

Thus, every regular solution to (X.0.8)

satisﬁes identity (X.1.1) for all ϕ ∈

D(Ω). Conversely, as in the case where Ω is bounded (cf. Section IX.1), it

is easy to show by means of Lemma III.1.1 that if v is of class C

(Ω), say,

and satisﬁes (X.1.1) for some f ∈ C(Ω) and all ϕ ∈ D(Ω), then v obeys

(X.0.8)

for some p ∈ C

(Ω). However, if v merely satisﬁes (X.1.1) a nd is

not a priori suﬃciently regular, we cannot go from (X.1.1) to (X.0.8)

and

therefore (X.1.1) is the weak version of (X.0.8).

As in the linear case, we shall consider the more general situation when

the ri ght-hand side of (X.1.1) is deﬁned by a linear functional f ∈ D

−1,2

(Ω).

Thus, i n analogy with Deﬁnition V.1.1, we give the following.

Deﬁnition X.1.1. Let Ω b e an exterior domain of R

, n ≥ 2. A vector ﬁeld

v : Ω → R

is called a weak (or generalized) solution to the Navier–Stokes

problem (X.0.8), (X.0 .4) if and only if

Often, i n t he literature, generalized solutions are called D-solutions (or Leray

solutions), where D means that they have a ﬁnite Dirichlet i ntegral. We shall

avoid this nomenclature in the present chapter, while in Chapter XII, which

treats the two-dimensional case, it will be used to mean a ﬁeld satisfying the

requirements (i)-(iii) and (v) of Deﬁnition X.1.1 but not necessarily condition

(iv). I t should be emphasized that in three dimensions any such ﬁeld obeys (iv)

for some v

∞

∈ R

, as a consequence of Theorem II.6.1.

654 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

(i) v ∈ D

1,2

(Ω);

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

(iii) v satisﬁes the boundary condition (X.0.8)

(in the trace sense) or, if

∗

≡ 0, then ϑv ∈ D

1,2

(Ω) where ϑ ∈ C

(Ω) and ϑ(x) = 1 if x ∈ Ω

R/2

and ϑ(x) = 0 if x ∈ Ω

, R > 2δ(Ω

);

(iv) lim

|x|→∞

n−1

|v(x) + v

∞

| = 0;

(v) v satisﬁes the identity

(∇v, ∇ϕ) + R(v ·∇v, ϕ) = −R[f, ϕ] (X.1.2)

for all ϕ ∈ D(Ω).

Remark X.1.1 Remark V.1.1 with q = 2, and Remark IX. 1.1, Remark

IX.1. 3 equal ly apply to generalized solutions of Deﬁnitio n X.1.1. 

Our next objective is to ascertain the existence of a pressure ﬁeld associ-

ated to a generalized solution and to investigate the corresponding properties.

Existence is, in fa ct, readily established. Actually, if

f ∈ W

−1,2

(Ω

), for every bounded domain Ω

with Ω

⊂ Ω,

from Lemma II.6.1 and Lemma IX.1.2 it follows tha t there is a ﬁeld

p ∈ L

loc

(Ω)

verifying the identity

(∇v, ∇ψ) + R(v · ∇v, ψ) = (p, ∇·ψ) − R[f , ψ] (X.1 .3)

for a ll ψ ∈ C

∞

(Ω). In particular, if

f ∈ W

−1,2

(Ω

and Ω is locally Lipschitz, again from Lemma II.6.1 and Lemma IX.1.2, we

deduce

p ∈ L

(Ω

However, unlike the case where Ω is bounded, we are not able a priori to draw

any conclusion concerning the L

-summability of p on the whole of Ω. This is

due to the circumstance that if we only have

v ∈ D

1,2

(Ω)

we cannot guarantee the existence of c = c(v) such that

|(v · ∇v, ψ)| ≤ c|ψ|

1,2

. (X.1.4)

Nevertheless, if we restrict ourselves to three-dimensional ﬂows, we can show

summability for p with a suitable exponent in a neighborhood of inﬁnity. In

this regard, we can show the following.

X.1 Generalized Solutions and Related Properties 655

Lemma X.1.1 Let Ω be an exterior domain of R

, n = 2, 3, and let v be a

generalized solution to (X.0.8), (X.0.4) in Ω. Then, if

f ∈ W

−1,2

(Ω

) (X.1.5)

for every bounded domain Ω

with Ω

⊂ Ω, there exists

p ∈ L

loc

(Ω)

satisfying (X.1.3) for all ψ ∈ C

∞

(Ω). If Ω is locall y Lipschitz and for some

R > δ(Ω

)

f ∈ W

−1,2

(Ω

we have

p ∈ L

(Ω

Moreover, if Ω ⊆ R

and, in addition to (X.1.5),

f ∈ D

−1,q

(Ω

), some R > δ(Ω

), and q ∈ (3/2, ∞),

then p (up to an additive constant) admits the following decomposition

p = p

+ p

, p

∈ L

(Ω

) , p

∈ L

(Ω

) , ρ > R. (X.1.6)

For this latter results to hold, no regularity o n Ω is needed.

Proof. By what we a lready said, we have to prove only (X.1.6). We shall do

this under the assumption v

∞

= 0, the case v

∞

6= 0 being treated i n an

entirely anal ogous way. For i = 1, 2, let

, p

) ∈ D

1,q

(Ω

) × L

(Ω

), q

= 3 , q

= q ,

be q

-generalized solution and asso ciated pressure to the foll owing Stokes prob-

lems

(∇v

, ∇ψ) = (p

, ∇ ·ψ) + [f

, ψ] , ψ ∈ C

∞

(Ω

) ,

:= −Rv ·∇v , f

:= −Rf .

(X.1.7)

Since by (iv) of Deﬁnition X.1.1 and Theorem II.6.1 we have

v ∈ L

(Ω

)

and therefore, for all ψ ∈ C

∞

(Ω

|(v · ∇v, ψ)| = |(v ⊗ v, ∇ ψ)| ≤ kvk

6,Ω

|ψ|

1,3/2

we conclude that

∈ D

−1,3

(Ω

By this and by the assumption on f, with the help of Theorem V.5.1

show the existence of the above solutions. Next, from (X.1.7) and (X.1.3), it

follows that

And of Theorem VII.7.2 and Remark VII.7.3 when v

∞

6= 0 .