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

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

along with the side condition

lim

|x|→∞

(v(x) + v

∞

) = 0 , v

∞

:= v

(e · e

+ ω e

× x. (XI.0.2)

Our investigation of problem (XI.0.1)–(XI.0.2 ) will be limited to the three-

dimensional case only, in that no general results are availabl e, to da te, for a

two-dimensional domain, with the exception of very special cases.

Further-

more, for the sake of simpli ci ty, we assume that

◦

Ω

is connected .

The fundamental challenge with (XI.0.1)–(XI.0.2) consists in the fact that

the velocity ﬁeld becomes unbounded at large distances from ∂Ω. An immedi-

ate consequence of this circumstance is that probl em (XI.0.1)–(XI.0.2) cannot

by any means b e viewed as a perturbation to the analogous problem with

ω = 0, which we analyzed in the previous chapter.

Notwithstanding this diﬃculty, thanks to the remarkable fact that the

total power of the “rotational term”

Ω

× x · ∇ u − e

× u) · u vanishes

identicall y along diﬀerentiable vector functions u compactly supported in Ω,

one can prove that all solutions (in a suitable class) to problem (XI.0.1)–

(XI.0.2) satisfy an a priori estimate analo gous to (X.0.5), that is,

Ω

∇(v + v

∞

) : ∇(v + v

∞

) ≤ M , (XI.0.3)

with M depending only on the data. As a consequence, by appropriately

modifying the procedure used for the proof o f Theorem X.4.1, we can prove the

existence of a generalized solution, that is, D-solutions, for data of arbitrary

“size,” also in the case ω 6= 0. This fact was ﬁrst pointed out by Leray (1933,

Chapter III); see also Borchers (1992, Korollar 4 .1).

Now, though it is a simple job to show that to every generalized solution v

one can associate a locally integrable pressure ﬁeld p and that the pair (v, p)

is smooth (for as long as the data and the domain may allow), the question

whether a generali zed solution is also physically reasonable appears to be a

much more challenging task. In accordance to what we discussed in the In-

troduction to the previous chapter, by “physically reasonable” (PR) we mean

a solution that satisﬁes the energy equation (X.0.6), that for “small” data

is unique, and that mo reover, under suitable circumstances, shows a “wake-

like” behavior, namely, a region outside which the velocity ﬁeld converges to

its asymptotic value much more rapidly than inside. As we know from the

study o f the case ω = 0, the proof of all these properties can be achieved if

a solution has “good” behavior at large distances. However, for a D-solution,

in addition to (XI.0.3), the only other inf ormation we have at the outset on

its asymptotic behavior is

Ω

|v + v

∞

≤ M

, (XI.0.4)

For example, if Ω is the exterior of a rotating circle, the solution assumes the

very simple form given in (V.0.8)

, (V.0.9).

XI Introduction 747

as a consequence of (XI.0.3) and Theorem II.6.1.

The question of existence of PR solutions was initiated by Ga ldi (2003)

for the case R

= 0 (rotation without translation, that is), and continued and,

to some extent, compl eted in the general case, by Galdi & Silvestre (2007a,

2007b). Speciﬁcally, by an entirely diﬀerent approach from that one adopted

by Finn (1965a) fo r the case ω = 0, in those papers it is proved that if the

data f and v

∗

are “small” in a suitable sense, then problem (XI.0.1)–(XI.0.2)

possesses one (and, in fact, only one) PR solution. As in the case ω = 0, while

it is immediate to show that these solutions are also D-soluti ons, the converse

property i s in no way obvious, even for “small” data.

The probl em of whether a D-solution is also physically reasonable has

been recently investigated by Galdi & Kyed (2010, 2011a). Their main results

heavily rely on the analysis of the generalized Oseen problem performed in

Chapter VIII, and can be summarized as follows.

Suppose, at ﬁrst, v

6= 0 and e ·e

6= 0 , namely,

· ω 6= 0 , (XI.0.5)

and let (v, p) be a solution to (XI.0.1) with f m ildly regular and of bounded

support,

and v satisfying (XI.0.3), (XI.0.4). Then, there exists a semi-inﬁnite

cone, C, whose axis is directed along the ±e

-direction according to whether

· ω

0, and such that for all suﬃciently large |x| and δ > 0,

v(x) + v

∞

(x) =

(

O(|x|

−1

) uniformly ,

O(|x|

−3/2+δ

) if x 6∈ C,

∇(v(x) + v

∞

(x)) =

(

O(|x|

−3/2

) uniformly ,

O(|x|

−2+δ

) i f x 6∈ C .

(XI.0.6)

Furthermore, there is p

∈ R, such that

ep(x) − p

= O(|x|

−2

ln |x|) , (XI.0.7)

where ep(x) = p(x) +

+ x

Notice that the asymptotic properties

given in (XI.0.6) coincide with those of the same quantity in the case ω = 0

and v

6= 0, given in Theorem X.8.1 a nd Theorem X.8.2. However, the main,

and in principle substantial, diﬀerence between the cases ω = 0 and ω 6= 0

relies on the fact that in the rotational case it is not known, to date, whether

there existence a leading term in the asymptotic behavior of the velocity

ﬁeld, while in absence of rotation we proved that such a term exists and

coincides with the Oseen fundamental tensor E, in the sense speciﬁed in

Theorem X.8.1 and Theorem X.8.2. As a consequence, it is not known whether

This latter assumption can be fairly weakened, by imposing only that f decays

“suﬃciently fast” at large distances.

Observe that the ﬁeld ep is, in fact, the real pressure of the liquid; see Section I.2.

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

the asymptotic estimates given in (XI.0.6) are optimal. In particular, it is not

known whether the kinetic energy of the liquid when v

∗

≡ f ≡ 0 is inﬁnite,

as in the irrotational case, or else is ﬁnite.

It is interesting to give an interpretation of the above results and of condi-

tion (XI.0.5), in the case in which v

is the translational velocity of the center

of mass G and ω is the angular velocity of the “body” B ≡ Ω

, moving in a

viscous liquid that is quiescent at spatial inﬁnity. In such a case, we recall

that condition (XI.0.5) ensures that with respect to an inertial frame I, the

velocity η of G has a nonzero component η

in the direction e

of ω. On

physical g rounds, we thus expect the formation of a wake region behind the

body i n the direction opposite to η

. The cone C is then exactly representative

of this wake region. We al so would like to emphasize that, again on physical

grounds, condition (XI.0.5) is necessary for the formation of the wake. In fact,

if v

·ω = 0, then the motion of B in I reduces to a pure rotation

where, of

course, no wake region is expected.

In the process of proving the estimates (XI.0.6) and (XI.0.7) we al so ﬁnd,

on the one hand, that every D-solution is unique in its own class if the data

are “suﬃciently small,” and on the other hand, that every D-solution satisﬁes

the energy equation.

If v

= 0 or e · e

= 0, namely,

· ω = 0 , (XI.0.8)

the picture is less clear, and the corresponding results resemble those found

in the previous chapter when v

∞

= 0. More precisely, following Galdi &

Kyed (2010), we show that every D-solution satisfyi ng the energy inequality,

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

and corresponding to “suﬃciently

small” data, has asymptotic behavior similar to that of the fundamental Stokes

solution. More precisely,

(v(x)+ω ×x) = O(|x|

−|α|−1

) , D

(ep(x)−p

) = O(|x|

−|α|−2

) , |α| = 0, 1 ,

(XI.0.9)

for some p

∈ R. The proof of (XI.0.9) is similar, in principle, to that of

Theorem X.9.1 in the case ω = 0. However, unlike this latter, the proof of

the appropriate summability properties of the pressure ﬁeld associated to a

D-solution is quite elaborate.

The question whether the above results continue to hold for data of “ar-

bitrary” size, as in the case v

·ω = 0, remains open. However, on the bright

This happens, for example, in the very important situation of a rigid body B

translating and rotating in a viscous liquid when the walls of B are ﬁxed and

impermeable; see also the comments in the next paragraph.

See footnote 14 of Chapter I.

Under more general assumptions on the body force f ; see Theorem XI.5.1 and

Theorem XI.5.3.

This class is proved to be not empty.

XI.1 Generalized Solutions and Related Properties 749

side, unlike the case v

· ω 6= 0 , for the case at hand, Fa rwig, Galdi & Kyed

(2010), relying on the previous work of Galdi (2003) and Farwig a nd Hishida

(2009), were able to show, for small data at least, the existence of a leading

term in the pointwise asymptotic behavior of a generalized solution. In par-

ticular, regarding the velocity, one shows that it coincides with the velocity

ﬁeld of a suitable Landau solution.

We conclude this introductory section by rewriting (XI.0.1), (XI.0.2) in a

suitable nondimensional form. If v

·ω 6= 0, we scale the velocity by v

e·e

and

the length by d := δ(Ω

), so that by a simple computation, (XI.0.1) becomes

∆v = Rv · ∇v + 2T e

×v + ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω ,

(XI.0.10)

where R and T are deﬁned in (VIII.0.6)

and (VIII.0.3)

, respectively. If, on

the other hand, v

·ω = 0, then we scale the velocity with ω d, whi ch amounts

to setting (formal ly) in (XI.0.10) R = T . As far as (XI.0.2) is concerned, we

have

lim

|x|→∞

(v(x) + v

∞

(x)) = 0 , v

∞

(x) :=

(

× x , if v

· ω 6= 0 ,

×x , if v

· ω = 0 .

(XI.0.11)

Throughout this chapter, we will focus on the study of the boundary-value

problem (XI.0.10)–(XI.0.11).

XI.1 Generalized Solutions. Existence of t he Pressure

and Regularity Properties

The weak formulation o f the boundary-value problem (XI.0.10), (XI.0.11) is

given with a by now standard procedure that is the natural generalization of

its irrotational counterpart. Speciﬁcally, if we formal ly multiply (XI.0.10 )

ϕ ∈ D(Ω) and integrate by parts over Ω, we obtain

Ω

∇v : ∇ϕ + R

Ω

v · ∇v · ϕ + 2T e

Ω

v ·ϕ = −

Ω

f · ϕ. (XI.1.1)

As usual, we shall consider the more general situation in which the right-

hand side of (XI.1.1) is deﬁned by a linear functional f ∈ D

−1,2

(Ω).

Thus, in complete analogy w ith Deﬁnition X.1.1 (and Deﬁnition VIII.1.1),

we give the following.

Deﬁnition XI.1.1. Let Ω be an exterior dom ain of R

A vector ﬁeld v :

Ω → R

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

(XI.0.10), (XI.0. 11) if

See Remark X.9.3 for the deﬁnition of a Landau solution.

We recall that we will assume throughout that Ω

is connected.

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

(i) v ∈ D

1,2

(Ω);

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

(iii) v satisﬁes the boundary condition (XI.0.10)

(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|→∞

|v(x) + v

∞

| = 0 ;

(v) v satisﬁes the identity

(∇v, ∇ϕ) + R(v · ∇v, ϕ) + 2T (e

×v, ϕ) = −[f, ϕ] (XI.1.2)

for all ϕ ∈ D(Ω).

Remark XI.1.1 R emark V.1.1 with q = 2, and Remark IX.1.1 equally apply

to generalized solutions of Deﬁnition XI.1.1. 

We shall next investigate the existence and the properties of the pressure

ﬁeld associated to a generalized solution. W hile existence is a simple task,

the proof of global summability properties requires more eﬀort, and heavily

relies on the properties of weak solutions to the generalized Oseen problem

established in Section VIII.2. We begin with the following.

Lemma XI.1.1 Let v be a generalized solution to (XI.0.10), (XI.0.11). Then,

f ∈ W

−1,2

(Ω

) (XI.1 .3)

for every bounded domain Ω

with Ω

⊂ Ω, there exists

p ∈ L

loc

(Ω)

such that

(∇v, ∇ψ) + R(v · ∇v, ψ) + 2T (e

× v, ψ) = (p, ∇ · ψ) − [f, ψ] (XI.1.4)

for all ψ ∈ C

∞

(Ω). Furthermore, if Ω is locally L ipschitz and for some R >

δ(Ω

f ∈ W

−1,2

(Ω

then we have

p ∈ L

(Ω

Proof. The proof of the lemma is completely analogous to that given in

Lemma X.1.1 and will be therefore left to the reader. ut

Lemma XI.1.2 Let v and f be as in Lemma XI.1.1. Suppose, in addition,

f ∈ L

(Ω

) . for some ρ > δ(Ω

Then v ∈ D

2,2

(Ω

), for all r > ρ.

XI.1 Generalized Solutions and Related Properties 751

Proof. Under the stated assumptions on f , in the following Theorem XI.1.2 it

is shown that (v, p) ∈ W

2,2

loc

(Ω

) ×W

1,2

loc

(Ω

). Therefore, setting u := v + v

∞

from (XI.1.4), we obtain

∆u + R

∂u

∂x

+ T (e

×x · ∇u −e

× u) = Ru · ∇u + ∇ep + f

∇ · u = 0











a.e. in Ω

(XI.1.5)

where

ep :=

(

p +

+ x

), if v

·ω 6= 0

p +

+ x

), if v

· ω = 0 .

(XI.1.6)

We have to show that

u (≡ D

v) ∈ L

(Ω

) . (XI.1.7)

Let ψ

be the “cut-oﬀ” function used in the proof of Lemma VIII.2.1. By

dot-multiplying both sides of (XI.1.5)

by −∇×(ψ

∇×u), integrating over

Ω

, and taking i nto account (VIII.2.4), we derive

√

∆uk

= −(R

∂u

∂x

+ f, ψ

∆u + (∇ × u) × ∇ψ

)

−

((∇× u) × ∇(

√

∆u)

+T (e

×u, ∇ ×(ψ

∇× u) − (e

× x · ∇u, ∇×(ψ

∇ × u))

+R(u · ∇u, ψ

∆u + (∇ × u) × ∇ψ

) .

With the help of (VIII.2.6), (VIII.2.7), (VIII.2.9) and (VIII.2.11), from this

relation we deduce

∆uk

≤ c (kfk

+|u|

1,2

)+R(u·∇u, ψ

∆u+(∇×u)×∇ψ

) , (XI.1.8)

with c independent of R. We now estimate the last two terms on the right-hand

side of (XI.1.8). To this end, we begin by observing that since u belongs to

1,2

(Ω) and satisﬁes condition (iv) in Deﬁnition XI.1.1, by Theorem II.6.1(i)

it follows that u ∈ L

(Ω) and

kuk

≤ c

|u|

1,2

, (XI.1.9)

where c

= c

(Ω). By the H¨older inequality and (XI.1.9), we have

See footnote 3 in the Introduction to this chapter.

This formal calculation can be easily made rigorous by a simple approximation

procedure and by the use of the results of Exercise II.4.3. For simplicity, through-

out the proof, we set k · k

2,Ω

≡ k · k

, and (· , ·)

Ω

≡ (·, ·).

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

(u ·∇u, ψ

∆u) ≤ k

√

∇uk

kuk

√

∆uk

= k∇(

√

u) − u ⊗ ∇

√

kuk

√

∆uk

≤ c



k∇(

√

u)k

+ ku ⊗ ∇

√



|u|

1,2

√

∆uk

(XI.1.10)

From Nirenberg’s Lemma II.3.3 and Exercise II.7.4, we also obtain

k∇(

√

u)k

≤ c

k∇(

√

u)k

1/2

(

√

u)k

1/2

= c

k∇(

√

u)k

1/2

k∆(

√

u)k

1/2

(XI.1.11)

We next observe that by the properties of the function ψ

, (XI.1.9), and again

the H¨older inequality, we easily obtain

k∆(

√

u)k

≤ k

√

∆uk

+ 2k∇u · ∇

√

+ ku∆

√

≤ k

√

∆uk

+ c

|u|

1,2

+ c

ρ,r

|u|

R,2R

|u|

1/2

≤ k

√

∆uk

+ c

(|u|

1,2

+ kuk

)

≤ k

√

∆uk

+ c

|u|

1,2

(XI.1.12)

with c

independent of R. By a similar argument, we also obtain

k∇(

√

u)k

≤ c

ρ,r

|u|

R,2R

|u|

1/2

+ |u|

1,2

≤ c

(kuk

+ |u|

1,2

) ≤ c

|u|

1,2

(XI.1.13)

with c

independent of R. Thus, i nserting the above inequal ities into (XI.1.11),

we deduce

k∇(

u)k

≤ c



|u|

1/2

1,2

∆uk

1/2

+ |u|

1,2



Since by an argument similar to that leading to (XI.1.13) we get

ku ⊗ ∇

≤ c

ρ,r

|u|

R,2R

|u|

1/2

≤ c

kuk

≤ c

|u|

1,2

from (XI.1.10) we deduce

(u · ∇u, ψ

∆u) ≤ c



|u|

1/2

1,2

∆uk

1/2

+ |u|

1,2



|u|

1,2

∆uk

If we use Young’s inequality (II.2.5) in this relation, we ﬁnally conclude that

(u · ∇u, ψ

∆u) ≤ c



|u|

1,2

+ |u|

1,2



+ εk

∆uk

, (XI.1.14)

XI.1 Generalized Solutions and Related Properties 753

where ε > 0 is arbitrary and c

depends on ε, but is otherwise independent

of R. It remains to estimate the last term on the right-hand side of (XI.1.8).

Using one more time the H¨older inequality, (XI.1.9), and Nirenberg’s Lemma

II.3.3, we deduce

(u · ∇u, (∇ × u) × ∇ψ

) ≤ c

kuk

|u|

1,2

√

∇uk

≤ c

|u|

1,2

√

∇uk

1/2

k∇(

√

∇u)k

1/2

≤ c

|u|

5/2

1,2



(

√

u)k

+ kuD

√



1/2

Thus, by (XI.1.12), (XI.1. 9), the properties of ψ

, and Young’s inequality,

from the previous inequality we obtain

(u · ∇u, (∇ × u) × ∇ψ

) ≤ c



|u|

5/2

1,2

√

∆uk

1/2

+ |u|

1,2



≤ c



|u|

10/3

1,2

+ |u|

1,2



+ εk

√

∆uk

(XI.1.15)

where ε > 0 is arbitrary and c

depends on ε, but is otherwise independent

of R. Collecting (XI.1.8), (XI.1 .14), and (XI.1.15), and choosing ε suﬃciently

small, we conclude that k

√

∆uk

≤ M , with a constant M independent of

R. This inequality formally coincides with (VIII.2.12 ), so that, arguing as in

the proof of Lemma VIII.2.1, we prove (XI.1.7). ut

The previo us lemma has the following interesting consequence.

Corollary XI.1.1 Let v be a generalized solution to (XI.0.10), (XI.0.11)

corresponding to f that satisﬁes the assumptio n of Lemma XI.1.2. Then,

v ∈ L

∞

(Ω

), r > ρ, and moreover,

lim

|x|→∞

(v(x) + v

∞

(x)) = 0 , uniformly. (XI.1.16)

Proof. Under the given assumptions on f, in Theorem XI.1.2 below it will be

proved that v ∈ W

2,2

(Ω

r,R

), for all R > r, and so by the embedding Theorem

II.3.4, v + v

∞

∈ L

∞

(Ω

r,R

), for all R > r. Consequently, in order to prove the

corollary, it is enough to prove (XI.1.16). Since v is a generalized solution (see

(XI.1.9)), we know that

u ∈ L

(Ω

) ∩D

1,2

(Ω

) , (XI.1.17)

where u := v +v

∞

. Moreover, from Lemma XI.1.2 we also have u ∈ D

2,2

(Ω

which, in turn, by (XI.1.17) and Theorem II.6.1, implies u ∈ D

1,6

(Ω

The property (XI.1.16) then follows from this l atter, (XI.1.17), and Theorem

II.9.1 .

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

We are now in a position to prove the global summability properties of

the pressure.

Theorem XI.1.1 Let v be a generalized solution to (XI.0. 10), (XI.0.11), and

let f satisfy (XI.1.4) and, in addition,

f ∈ L

(Ω

) ∩ L

(Ω

) , some q ∈ (1, ∞) and ρ > δ(Ω

) .

Let p be the pressure ﬁeld associated to v by Lemma XI.1.1, with ep deﬁned

in (XI.1.6). Then ep, possibly modiﬁed by the addition of a constant, admits

the following decomposition:

ep = p

+ p

, (XI.1.18)

where

∈ L

(Ω

) ∩ D

1,2

(Ω

) ∩ D

1,q

(Ω

) , p

∈ L

(Ω

) , r > ρ. (XI.1.19)

Moreover, if q ∈ (1, 3), we also have

∈ L

3q/(3−q)

(Ω

) . (XI.1.20)

Thus, i n particular, if q ∈ (1, 3/2), we conclude that

ep ∈ L

(Ω

) . (XI.1.21)

Proof. We set u := v +v

∞

, and recall that, by Theorem XI.1.2 , stated below,

and the assumptions on f , we can take (u, ep) ∈ W

2,2

loc

(Ω

) × W

1,2

loc

(Ω

), so

that in particular, (u, ep) satisﬁes (XI.1 .5). Therefore, we obtain

∆u + R

∂u

∂x

+ T (e

×x · ∇u −e

× u) = ∇ep + f + ∇·(u ⊗u)

∇ · u = 0











a.e. in Ω

(XI.1.22)

We next notice that by (XI.1 .9),

u ⊗ u ∈ L

(Ω

) .

This latter property, along with the assumption on f , allows us to conclude, by

Lemma VIII.2.2, the validity of (XI.1.18 )–(XI.1.21). The proof of the theorem

is complete. ut

Remark XI.1.2 We notice that in view of Remark VIII.2.1, Theorem XI.1.1

continues to hold if T = 0. 

We conclude this section by summarizing, in the next theorem, the diﬀer-

entiability properties of generalized solutions. Their proof is entirely analo gous

(and simpler, since we are in three dimensions) to that of Theorem X.1.1, and

we leave it to the reader.

XI.2 On the Energy Equation and the Uniqueness of Generalized Solutions 755

Theorem XI.1.2 Let v be a generali zed solution to (XI.0.10), (XI.0.11).

Then, if

f ∈ W

m,q

loc

(Ω), m ≥ 0 , (XI.1.23)

where q ∈ (1, ∞) if m = 0, whil e q ∈ [3/2, ∞) if m > 0, it follows that

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω),

where p is the pressure associated to v by Lemma XI.1.1. Thus, in particular,

f ∈ C

∞

(Ω),

then

v, p ∈ C

∞

(Ω).

Assume further that Ω is of class C

m+2

and

∗

∈ W

m+2−1/q,q

(∂Ω), f ∈ W

m,q

(Ω

) ,

for some R > δ(Ω

) and with the values of m and q speciﬁed earlier. Then,

we have

v ∈ W

m+2,q

(Ω

), p ∈ W

m+1,q

(Ω

Therefore, i n particular, if Ω is of class C

∞

and

∗

∈ C

∞

(∂Ω), f ∈ C

∞

(Ω

it follows that

v, p ∈ C

∞

(Ω

XI.2 On the Energy Equation and t he Uniqueness of

Generalized Soluti ons

In this section we shall investigate two important properties of generalized

solutions that, a s shown in Sections X.2 and X.3 in the irrotational case,

may be somewhat related. We will begin by gi ving suﬃcient conditions for a

generalized solution to satisfy the energy equation:

Ω

D(v) : D(v) −

∂Ω



∞

+ v

∗

) · T (v, ep) −

∗

+ v

∞

)

∗



· n

Ω

f · (v + v

∞

) = 0 ,

(XI.2.1)

where v

∞

is deﬁned in (XI.0.11), ep in (XI. 1.6), and T is the Cauchy stress

tensor (IV.8.6). This relation is fo rmally obtained by dot-multiplying both

sides of (XI.0.10) by v + v

∞

, integrating over Ω

, and assuming that all the