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

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576 VIII Steady Generalized Oseen Flow in Exterior Domains

Finally, suppose that for som e r ∈ (1, ∞) and al l R > δ(Ω

), (v

, p

) ∈

2,r

(Ω

) × W

1,r

(Ω

) is another solution to (VIII.0.7) corresponding to the

same data and that in addi tion,

v ∈ L

(Ω) , for some q ∈ (1 , ∞) , (VIII.8 .12)

Then v(x) = v

(x) , p(x) = p

(x) + p

for a .a. x ∈ Ω, and some constant p

Proof. We begin by observing that the uniqueness part is an immediate con-

sequence of Theorem VI II.7.1. It remains to prove existence. For given f and

∗

satisfying the assumptions of the theorem, we let {f

} and {v

∗k

} be se-

quences of smooth functions, with f

of bounded support for each k ∈ N,

converging to f and v

∗

in the L

- and W

2−1/q,q

-norms, respectively. We next

denote by v

, k ∈ N, the (unique) generalized solution corresponding to f

∗k

, and by p

∈ W

1,2

(Ω

) the associated pressure ﬁeld. We also set

= ψ (v

+ Φ

σ) + w

, φ

= ψ p

where ψ, Φ

, σ, and w

are the same quantities introduced in the proof of

Theorem VIII.7.2. We next observe that u

and φ

satisfy (VIII.5.1)

1,2

with

f ≡ F

, where

= f

+ ∆w

+ R

∂w

∂x

+ T



× w

−e

× w



+Rv

∂ψ

∂x

−p

∇ψ + 2∇ψ · ∇v

+ v

∆ψ

+2Φ

∇ψ · ∇σ + Φ

σ∆ψ + RΦ

∂ψ

∂x

where we have used (VIII.1.8). Therefore, recalling (VIII.7.12)

and that for

all R > δ(Ω

∈ W

2,2

(Ω

) ∩ L

(Ω) , p

∈ W

1,2

(Ω

) (VIII.8.13)

(see Remark VIII.1.1) we show, as in the proof of Theorem VIII.7.2, that

≤ C (kf

+ kv

1,q,Ω

2σ

+ kpk

q,Ω

2σ

) , for all q ∈ (1, 2), (VIII.8.14)

where C = C(σ, Ω, q, B, T). Now again by (VIII.8.13), we have

∈ L

) ∩ W

2,2

) , Φ

∈ W

1,2

) ,

for all R > 0, and so by Lemma VIII.8.2 and Lemma VIII.7.1 a nd (VII I.8.14),

we obtain, on the one hand, that (u

, φ

) is in the class C

), for all k ∈

N, and that on the other ha nd, it must satisfy (VIII. 8.4)–(VIII.8.7) with f

replaced by F

. Since u

(x) = v

(x), φ

(x) = p

(x) for a ll x ∈ Ω

2σ

, by

(VIII.8.14) we thus obta in

VIII.8 Existence, Uniqueness, and L

-Estimates. The Case R 6= 0 577

2,q,Ω

2σ

+ R



∂v

∂x



q,Ω

2σ

+ T ke

×x ·∇v

−e

× v

q,Ω

2σ

1/4

4q/(4−q),Ω

2σ

1/2

2q/(2−q),Ω

2σ

+|p

1,q,Ω

2σ

+kp

3q/(3−q),Ω

2σ

≤ c



1 +



(kf

+ kv

1,q ,Ω

2σ

+ kp

q,Ω

2σ

) ,

(VIII.8.15)

with c

= c

(σ, q, Ω, B, T). However, by Lemma VIII.6 .1, we also have

2,q,Ω

2σ

+kp

1,q,Ω

2σ

≤ c

(kf

q,Ω

3σ

+ kv

∗k

2−1/q,q,∂Ω

+ kv

q,Ω

3σ

+ kp

q,Ω

3σ

(VIII.8.16)

where c

= c

(Ω, σ, B, T). Therefore, combining (VIII.8.15) and (VIII.8.16),

we deduce

2,q ,Ω



∂v

∂x



q,Ω

+ T ke

×x ·∇v

−e

× v

q,Ω

1/4

4q/(4−q),Ω

+ R

1/2

2q/(2−q),Ω

+ |p

1,q ,Ω

+ kp

3q/(3−q),Ω

≤ c



+ kv

∗k

2−1/q,q,∂Ω

+ kv

1,q,Ω

3σ

+ kp

q,Ω

3σ



(VIII.8.17)

Using a by now standard contradiction argument already employed several

times previously, and which relies o n the embedding W

1,q

(Ω

) ,→,→ L

(Ω

)

(see Exercise II.5.8) and the uniqueness Theorem VIII.7.1, we prove the exis-

tence of a constant c

depending on Ω, σ, R, a nd T but otherwise i ndependent

of k ∈ N such that

1,q,Ω

3σ

+ kp

q,Ω

3σ

≤ c



+ kv

∗k

2−1/q,q,∂Ω



However, we observe that if q ∈ (1, 3/2), the constant c

may be taken de-

pendent only on q, Ω, B, and T , whenever R ∈ (0, B]. The argument that we

need to prove this assertion is entirely analogous to that given in the proof of

Theorem VII.7.1 , and precisely in that part following (VII.7.1 9). We leave it

to the reader. From this latter inequality and (VIII.8.17) we infer

2,q



∂v

∂x



+ T ke

× x · ∇v

− e

×v

1/4

4q/(4−q)

+ R

1/2

2q/(2−q)

+ |p

1,q,Ω

+ kp

3q/(3−q)

≤ c



+ kv

∗k

2−1/q,q,∂Ω



(VIII.8.18)

where for simplicity, we have omitted the subscript Ω. We now let k → ∞

in (VIII.8.18), and using the fact that f

→ f in L

(Ω) and v

∗k

→ v

∗

in W

2−1/q,q

(∂Ω), we easily establish, again by a standard argument, that

{(v

, p

)} tends, in the topology deﬁned by the norms on the left-hand side

578 VIII Steady Generalized Oseen Flow in Exterior Domains

of (VI II.8.18), to a pair (v, p) ∈ C

(Ω) satisfying (VIII.0.7), (VIII.0.2). The

proof of the theorem is thus complete. ut

Remark VI II.8.2 We wish to clarify how the sol utions of Theorem VIII.8.1

satisfy the condition at inﬁnity (VIII.0.2). If q ∈ (3/2, 2), since v ∈ D

2,q

(Ω),

by the embedding Theorem II.6.1(i) it follows that v ∈ D

1,q

∗

(Ω), for some

∗

> 3. Therefore, since v ∈ L

2q/(2−q)

(Ω), from Theorem II.9.1 we deduce

lim

|x|→∞

v(x) = 0 , uniformly po intwise.

If q ∈ (1, 3/2], in view of the fa ct that v ∈ D

1,4q/(4−q)

(Ω), it follows that

v ∈ D

1,s

(Ω), for some s ∈ (4/3, 12/5]. Consequently, from Lemma II.6.3 we

have

|v(|x|, ω)|dω = o



|x|

1−3/s



as |x| → ∞.



Theorem VIII.8.1 immediately leads to the fol lowing corollary.

Corollary VIII.8.1 Let Ω and R be as in Theorem VIII.8.1. Then, for any

f, v

∗

satisfying (VIII.8.10), problem (VIII.0.7), (VIII.0.2) has one and only

one corresponding solution (v, p) in the class C

(Ω). Mo reover, this soluti on

satisﬁes the inequality (VIII.8.11).

Finally, taking into account Remark VIII.1.1 and Theorem VIII.1.1, from

Theorem VIII.8.1 we also at once deduce the foll owing.

Corollary VIII.8.2 Let Ω and R be as in Theorem VIII.8.1, and let v

be a generalized solution to problem (VIII.0.7), (VIII.0.2). If f, v

∗

satisfy

(VIII.8.10), then v and its associated pressure p (by Lemma VIII.1.1) are in

the class C

(Ω). Moreover, (v, p) satisﬁes the inequality (VIII.8.11).

VIII.9 N otes for the Chapter

Section VIII.1. The ﬁrst result on the existence of generalized solutions to

(VIII.0.7), (VIII.0.2) can be traced back to the paper of Leray (1933, Chapter

III); see also Weinberger (1973 ), Serre (1987 ), and Borchers (1992, Korollar

4.1).

Theorem VIII.1.2 is due to me.

Existence, uniqueness, and corresponding estimates of q-generalized solu-

tions when R = 0 were proved by Hishida (2006) when Ω = R

and, more

generally, when Ω is a (smooth) exterior domain of R

. Hishida’s results are

formally analogous to those derived in Theorem IV. 2.2 and Theorem V.5.1 for

the Stokes problem, and in particular, if ∂Ω 6= ∅, they require the restriction

q ∈ (3/2, 3). In the case Ω = R

, Hishida’s results have been extended to

VIII.9 Notes for the Chapter 579

R 6= 0 by Kraˇcmar, Neˇcasov´a, & Penel (2006 ). See also Kraˇcmar, Neˇcasov´a,

& Penel (2 005, 2007, 201 0).

Weak solutions in Lorentz spaces have been studied by Farwig & Hishida

(2007) when R = 0. Their results are the analogous counterpart, for T 6= 0,

of those obtained by Kozono & Yamazaki (1998 ) for the Stokes problem. In

fact, they reduce to these latter when T = 0.

As we mentioned in the Introduction, there has been very little contribu-

tion to the study of the generalized Oseen problem for n = 2. As a matter of

fact, in two dimensions, the problem of existence of generalized solutions is

either rather well known or very complicated. Actually, because of n = 2, we

have to restrict ourselves to the case in which the angular velocity ω is per-

pendicular to the plane, Π, that contains the relevant region of motion of the

liquid. However, the translational velocity v

must belong to Π and therefore

is orthogonal to ω. Now, if ω = 0, we go back to the Oseen problem already

treated in great detail and solved in the previous chapter. If, however, ω 6= 0,

the Mozzi–Chasles transformation reduces the problem, formally, to the study

of (VIII.0.7), (VIII.0.2) with R = 0 and with v = (v

, x

), v

, x

)),

p = p(x

, x

). For this latter, using the same method employed in the proof of

Theorem VIII.1.2, one can prove the existence of a vector ﬁeld v ∈ D

1,2

(Ω)

that satisﬁes (VIII.1.1) (with R = 0) along with properties (ii) and (iii) of

Deﬁnition VIII.1.1. However, as in the analogous Stokes problem considered

in Chapter V, with this informa tion alone one cannot ensure that the velocity

ﬁeld tends (even in a weak sense) to a prescribed limit at inﬁnity. In other

words, one cannot exclude the occurrence of a “Stokes paradox.” It is inter-

esting to observe that, in the case that Ω is the exterior of a circle and f ≡ 0,

∗

= e

× x, the problem has the explicit solution given in (V.0.8) (with

ω ≡ e

). This fact would suggest that a solution might exi sts also when Ω is

a more general (suﬃciently smooth) exterior domain and with more general

data, possibly satisfying a suitable compatibility conditi on, but no proof is,

to da te, available. We end these considerations by mentioning the paper by

Farwig, Krbec, & Neˇcasov´a (2008) in which existence is investigated in the

case Ω = R

Section VIII.2. All main results and, in particular, the uniqueness Theorem

VIII.2.1 are due to me. Seeming ly, this is the ﬁrst (and only one, to my

knowledge) uniqueness theorem of generalized solutions i n their own class.

The main tool is the proof that the pressure possesses suitable summability

properties in a neighborhood of inﬁnity (see Lemma VIII.2.2).

The “regularization” result for generalized solutions obtained in Lemma

VIII.2.1 is also new. However, the existence of a generalized solution satisfying

the properties stated in Lemma VIII.2.1, including the estimate (VIII.2.3),

can also be proved by the method of “invading domains” adopted by Sil vestre

(2004).

Section VIII.3. The pro ofs of the main results (Lemma VIII.3.1 , Lemma

VIII.3.4, and Lemma VIII.3.6 ) are taken (and expanded) from the work of

580 VIII Steady Generalized Oseen Flow in Exterior Domains

Galdi & Silvestre (2007a, 2007b). A diﬀerent proof of Lemma VI II.3.1 can

also be found in Mizumachi (1984, L emma 2).

We would like to emphasize that Lemma VIII.3.1 admits a lso of a two-

dimensional counterpart. Speciﬁcally, one could show that for R = 0, which,

as we observed previously, is the only relevant case for two dimensions, the

fundamental tensor solution, Γ (ξ, τ), sati sﬁes the following estimates

(ξ) ≡

∞

|Γ (ξ, τ )|dτ ≤ c |log |ξ||,

(ξ) ≡

∞

|∇

Γ (ξ, τ)|dτ ≤ c |ξ|

−1

ξ 6= 0 , (VIII.8.1)

which coincide w ith the estimates satisﬁed by the Stokes fundamental tensor

solution U in dimension 2 (see (IV.2.6 )).

Section VIII.4. Theorem VIII.4.1 is basically due to O.A. Ladyzhenskaya.

Its proof appears for the ﬁrst time in English in Chapter 4, §5, of the ﬁrst

edition, dated 1963, of Ladyzhenskaya (1969).

Lemma VIII.4.2, in its generality, seems to be new and is due to me.

Theorem VIII.4.4, which represents the main contribution of the section,

is taken from Galdi & Silvestre (2007b). In this regard, and in connection with

the remarks made in the Notes to Section VIII.1, we would like to emphasize

that the method used in the proof of Theorem VIII.4 .4 fails in dimension

n = 2. In fact, we recall that fo r n = 2, we have to consider only the case

R = 0, and in such a case, according to (VIII. 8.1), the function Γ

(ξ) is

bounded above by log |ξ|, for large |ξ|. This prevents us from proving the

existence of solutions to (VIII.4.1) that are spatially decaying, even for a

right-hand side f with compact support in the space variables.

Section VIII.5. The entire approach described in this section is due to Galdi

(2002) and was further developed by Galdi & Silvestre (2007a, 2007b). In

particular, Theorem VIII.5.1 is proved in Galdi & Silvestre (2007b).

Section VIII.6. Theorem VIII.6.1 and Theorem VIII.6.2 are improved ver-

sions of analogous results derived in Galdi & Silvestre (2007a, 2007b).

Section VIII.7. The uniqueness results proved in Lemma VIII.7.1 and The-

orem VIII.7.1 are due to me. They also appear, in a slightly diﬀerent form, i n

Galdi & Kyed (2011a).

Existence and associated estimates of solutions in homogeneous Sobolev

spaces corresponding to f in L

were ﬁrst proved by Hishida (1999a, 1999b))

for q = 2. Speciﬁcally, Hishida (1999b, Proposition 3.1) shows, with Ω = R

that fo r every f ∈ L

) there exists one solution to (VIII.0.7) belonging

to D

2,2

) and satisfying corresponding estimates. Hishida’s result was ex-

tended to the case R 6= 0 by Galdi (2002, Lemma 4.14), who also proves

uniqueness.

The ﬁrst complete treatment of the problem for arbitrary q ∈ (1, ∞) and

Ω = R

is due to Fa rwig, Hishida, & M¨uller (2004) (for both cases n = 2, 3).

VIII.9 Notes for the Chapter 581

For n = 3 their existence result, based on multiplier and Littlewood–Paley

theories, essentially coincides with that given in Lemma VIII.7.2 , which is,

however, established by a completely diﬀerent approach.

Theorem VIII.7.2 is due to me.

Section VIII.8. Existence and corresponding L

-estimates when Ω = R

were ﬁrst derived by Farwig (2006). It should be noted that, as shown in

the proof of Lemma VIII.8 .2, the estimates associated with the ﬁrst deriva-

tives of the velo city ﬁeld of solutions constructed by Farwig have the puzzling

(but, apparently, necessary) feature that the constant involved depends on

the (square of the) inverse of the Taylor number T . Therefore, it becomes

unbounded as T → 0, even though one can easily show that as T → 0, the

corresponding solutions tend, in fact, in a suitable sense, to the solutio n of

the probl em with T = 0. As proved in Farwig (2005), this feature can be re-

moved if the ﬁrst derivatives are estimated in suitable weighted homogeneous

Sobolev spaces.

Theorem VIII.8.1 is due to me.

Steady Navier–Stokes Flow in Bounded

Domains

Kα`ι ´ηγ ´απησαν o`ι ˝ανϑρωπoι

µeαλ λoν τ `o ϕeωs η τ `o σκ´oτoς.

Introduction

The objective of this and the following chapters is the study of steady motions

of a viscous incompressible ﬂuid described by the full nonlinear Navier–Stokes

system. In the present chapter we shall focus on the case where the region

of ﬂow, Ω, is bounded. More speciﬁcally, we shall analyze the bounda ry value

problem obta ined by coupling the following system

ν∆v = v · ∇v + ∇p + f

∇· v = 0

)

in Ω (IX.0.1)

with the adherence boundary condition:

v = v

∗

at ∂Ω, (IX.0.2)

where v

∗

a prescribed ﬁeld.

The ﬁrst and fundamental contribution to the resolution of (IX.0.1),

(IX.0.2) for Ω bounded, is due to F.K.G. Odqvist in 1930. The method used

by this author articulates into several steps. First, he transforms (IX.0.1),

(IX.0.2) into a (nonlinear) integral equation, by means of Green’s tensor G

associated to the corresponding linearized Stokes problem (see Section IV.6);

583G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,

584 IX Steady Navier–Stokes Flow in Bounded Domains

then he derives suitable estima tes for G and combines these with a successive

approximation techni que that, ﬁnally, produces existence. However, for this

procedure to work it is necessary to restrict appropriately the size of the data.

At the time when Odqvist derived his results, such a restriction was, in a

sense, expected from b oth the physical and the mathematical points of view.

In fact, on the one hand, it conﬁrmed the idea that the agreement between the

Navier–Stokes theory and experiment should hold only at “small” Reynolds

numbers.

On the other hand, the highly nonlinear character of the problem

enhanced the possibili ty of solving it only “locally.” Therefore, when in his

celebrated paper of 1933, Jean Leray proved existence of solutio ns to (IX.0.1),

(IX.0.2) w ithout restrictions on the size of the data, the result sounded ab-

solutely remarkable, since it was not predictable from known experimental

observation, nor was i t obvious from the structure of the equations. Leray’s

proof was ba sed on the discovery that every solution to (IX.0.1), (IX.0.2) for-

mally obeys the following a priori estimate, whatever the size of the data may

be:

Ω

∇v : ∇v ≤ M,

where M depends only on the data f , v

∗

and on Ω and ν (see Section IX.3

and Section IX.4). Such a unif orm bound along with the Odqvi st estimate for

Green’s tensor and with a new method for determining ﬁxed points of nonlin-

ear maps in Banach spaces

allowed Leray to show the stated result; see Leray

(1933, 1936 ). This outstanding achievement, however, presents an undesired

feature in the case when the boundary ∂Ω has more than one connected com-

ponent, say, Γ

, i = 1, ..., m. In such a case, the compatibility condition on the

velocity v

∗

at the boundary, required by the incompressibility of the ﬂuid, is

∂Ω

∗

·n =

i=1

∗

· n = 0, (IX.0.3)

with n unit outer normal to ∂Ω, while Leray’s argument works if the condition

∗

·n = 0 i = 1, ..., m (IX.0.4)

is imposed (see also Hopf 1941, 1957). Clearly, if m > 1, (IX.0.4 ) is stronger

than (IX.0.3) and, in particular, it does not allow for the presence of extended

“sinks” and “sources” into the region of ﬂow. The question of whether prob-

lem (IX.0.1), (IX.0.2) admits a solution only under the natural restriction

This view relies on the experimental evidence that “laminar motions” such as

Couette or Poiseuille ﬂows are observed only for small values of the Reynolds

number R (“large” vi scosity ν) even though, as mathematical solutions, they

exists for all values of R. As is known, this view has been modiﬁed in light of

the modern theory of bifurcation.

This method will lead to the celebrated Leray-Schauder theorem, see Leray &

Schauder (1934).

IX Introduction 585

(IX.0.3) was left out by Leray a nd it still is a fundamental open question in

the mathematical theory of the Navier–Stokes equations.

In this chapter we study existence, uniqueness, and regularity of solutions

to problem (IX.0.1), (IX.0.2) with Ω bounded.

Following Ladyzhenskaya (1959b), we intro duce a variational formulation

of (IX.0.1), (IX.0.2) and correspondi ng generalized (weak) solutions. T he tech-

nique we shall use for existence of generalized solutions is diﬀerent (and sim-

pler) than Leray’s and is based on a variant of the Galerkin method we used

for the linearized Oseen problem in Section VI I.2. This approach is due to Fu-

jita (1 961) and, independently, to Vorovich & Youdovich (1961); see also Finn

(1965b, §2.7 ). Moreover, in place of condition (IX.0.4), we prove existence

under the following weaker assumption on the ﬂux of v

∗

through Γ

i=1



∗

·n



< ν

where c

, i = 1, ..., m, are computable constants depending on the domain Ω;

see Galdi (1991).

Concerning uniqueness, we obtain, as expected, that generalized solutions

are unique only for “small” data (or, equivalently, for “large” viscosity ν). We

also give examples of non-uniqueness for “ large” data (or, equivalently, for

“small” ν). These examples are based on ideas of Youdovich (1967) and, to

the b est of our knowledge, they are the only ones so far explicitly given fo r

the steady-state Navier–Stokes problem in a bo unded domai n with adherence

boundary conditions.

Regularity of a generalized solution i s investigated by means of a general

technique which, in fact, allows us to study regularity of a much larger class

of solutions (in arbitrary dimension n ≥ 2).

Finally, we analyze the behavior of generalized solutions i n the limit of

large viscosity ν (small R eynolds number). Speciﬁcally, we shall show that,

under suitable regularity assumptions on the data, such solutio ns tend uni-

formly pointwise, as ν → ∞, to solutions of the Stokes problem correspo nding

to the same data.

As in the linearized approximations, also for the full nonl inear equations

we shall ma inly be concerned with the physically interesting cases when Ω

is a three-dimensional or (for plane ﬂow) two-dimensional domain. However,

we shall also explicitly remark if and in what form a result can be extended

to spatial dimension n > 3. In this respect, we wish to notice that a ll main

results we prove in this chapter for n = 2, 3 carry over (more or less simply)

See also Borchers and Pileckas (1994) and the more recent results of Kozono

and Yanagisawa (2009a, 2009b). For other results under the general assumption

(IX.0.3), we refer the reader to the Notes for this Chapter.

For non-uniqueness, and even bifurcation, of steady-state solutions under periodic

boundary conditions, we refer to Zeidler (1997, §§ 72.7–72.8), where a detailed

analysis of these phenomena is performed for t he so called Taylor-Couette ﬂow.