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

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

It remains to estimate D

. To this end, we split the integral in (XI.6.37)

into two contributions as follows:

(y, t) =

(y )

(y − z, t − τ )D

(z, τ)V

(z, τ)] dz dτ ,

(y)

(y − z, t − τ )[V

(z, τ)D

(z, τ)] dz dτ ,

:= I

+ I

(XI.6.40)

where we have used the condition ∇ · V (z, t) = 0 for suﬃciently large |z|.

Thanks to (XI. 6.35) and (XI. 6.36), we obtain

| ≤ c

|y|

−1

(y)

∞

|∇Γ (y − z, t)|dt dz ,

so that by Lemma VIII.3.3, we deduce

| ≤ c

|y|

−1

. (XI.6.41)

We next notice that by an integration by parts,

= −

(y)

(y − z, τ )[V

(z, t − τ )V

(z, t − τ )] dz dτ ,

∂B

(y)

(y − z, τ ) V

(z, t − τ )V

(z, t −τ)n

dz dτ

:= I

(1)

+ I

(2)

(XI.6.42)

Employing (XI.6.36) and (XI.6.32), we obtain

(2)

| ≤ c

|y|

−2

∂B

(y )

∞

|∇Γ (y − z, t)|dt ≤ c

|y|

−2

. (XI.6.43 )

In order to estimate I

(1)

, we split it as sum of three integrals, I

, I

, and

, over B

R/2

, B

2R,R/2

− B

(y), and B

, respectively, where R := |y|, is

suﬃciently large. We have

| ≤ c

R/2



sup

s≥0

|V (z, s)|





∞

Γ ( y − z, t)|dt



dz ,

and consequently, using Lemma VIII.3.3, (XI.6.29), and Lemma XI. 6.3, we

get

| ≤ c

R/2



sup

s≥0

|V (z, s)|



|y −z|

−2

(1 + s(y − z))

−2

≤ c

|y|

−2

Ω

R/2



sup

s≥0

|u(Q

(s) · z)|



≤ c

|y|

−2

XI.6 On the Asymptotic Structure of Generalized Solutions When v

·ω 6= 0 787

for a rbitrary p ositive η. Thus we conclude that

| ≤ c

|y|

−2+η

. (XI.6.44)

Likewise, we show

| ≤ c



sup

s≥0

|V (z, s)|



|y −z|

−2

(1 + s(y − z))

−2

≤ c

Ω



sup

s≥0

|u(Q

(s) ·z)|



|y − z|

−2

dz .

Therefore, observing that for z ∈ Ω

we have |y −z| ≥

|z|+ R −|y| =

|z|,

by the H¨older inequality we deduce, for arbitrary r ∈ (1, 3),

| ≤ c

Ω



sup

s≥0

|u(Q

(s) · z)|





∞

−

r−1

dρ



r−1

which, in turn, by Lemma XI.6.3, delivers

| ≤ c

|R|

−2+η

= c

|y|

−2+η

(XI.6.45)

with η := 3(r − 1)/r positive and arbitrarily cl ose to zero, since r may be

taken arbitrarily close to 1. Finally, from (XI. 6.36) and Lemma VIII.3.3, we

obtain

| ≤ c

|y|

−2

R/2,2R

(y )

|y −z|

(1 + s(y − z))

≤ c

|y|

−2

1,3R

(y )

|y −z|

(1 + s(y − z))

(XI.6.46)

Since by a simple a nd direct calculation one shows that

1,3R

(y)

|y −z|

(1 + s(y −z))

≤ c

ln R

with c

independent of R, from (XI.6.46 ) we get

| ≤ c

|y|

−2

ln R ≤ c

|y|

−2+η

(XI.6.47)

for arbi trarily small positive η. Thus, recalling that I

(1)

i=1

and col-

lecting (XI.6.40)–(XI.6.4 7), we conclude that

|∇S

(y)| ≤ c



|y|

−1

+ |y|

−2+η



, (XI.6. 48)

where we would like to emphasize that the term |y|

−1

comes from estimating

the integral I

(2)

occurring in (XI.6.40). If we combine (XI.6.2), (XI.6. 8), and

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

(XI.6.12) together with (XI.6.38), (XI.6.39), and (XI.6.48), we obta in, for

suﬃciently large |x|,

|∇u(x)| ≤ |∇w(x)|+ |Φ||∇σ(x)| ≤ |∇S(Q(t) ·x)| + c

|x|

−3

≤ c



|x|

−1

+ |x|

−3/2

(1 + s(x))

−3/2

+ |x|

−2+η

+ t

−3/4

|wk



for a ll η > 0 and all t > 0. Thus, letting t → ∞ in this relation furnishes

|∇u(x)| ≤ c



|x|

−1

+ |x|

−3/2

(1 + s(x))

−3/2

+ |x|

−2+η



. (XI.6.49)

This inequality provides, in particular, ∇u(x) = O(|x|

−1

), which translates

into |∇V (y, t)| ≤ c |y|

−1

with c independent of y and t. Plugging this informa-

tion ba ck into the estimate for I

(2)

allows us to deduce the improved bound

(2)

= O(|y|

−2

), a nd so, recalling that the term |x|

−1

in (XI.6.49) is due only

to the contribution from I

(2)

, we may now replace that term w ith |x|

−2

, and

the proof of the theorem is complete.

We conclude this section with the following result regarding the asymptotic

behavior of the pressure.

Theorem XI.6.3 Let the assumptions o f Theorem XI.6.1 be satisﬁed. Then,

for a ll suﬃciently large |x|,

p(x) +

+ x

) + p

= O(|x|

−2

ln |x|) ,

for som e p

∈ R.

Proof. As usual, we set u := v + v

∞

, and take the divergence of both sides

of (XI.1.5) (with f ≡ 0). Recalling that ∇ · (e

× x · ∇u − e

× u) = 0, we

obtain

∆ep = ∇ · G in Ω

∂ep

∂n

= g on ∂Ω

(XI.6.50)

where ep is deﬁned in (XI.1.6) and

G := Ru · ∇u ,

g := [∆u + R(D

u − u · ∇u) + T (e

×x · ∇u −e

× u)] · n |

∂Ω

(XI.6.51)

From Lemma II.9.1 we establish the following representation of ep(x), for all

x ∈ Ω

ρ,r

and all r > ρ:

ep(x) =

Ω

ρ,r

G(y) · ∇ E(x −y)dy +

∂Ω

ρ,r

E(x −y)G(y) · n dσ

−

∂B

E(x − y)g(y) dσ

−

∂B

E(x −y)

∂ep

∂n

(y) dσ

∂B

∂E

∂n

(x − y)ep(y) dσ

∂B

∂E

∂n

(x − y)ep(y) dσ

(XI.6.52)

XI.6 On the Asymptotic Structure of Generalized Solutions When v

·ω 6= 0 789

where E i s the Laplace f unda mental solution (II.9.1). Using the prop erty

E(ξ)| ≤ c

|ξ|

−1−|α|

, |α| ≥ 0 , ξ ∈ R

−{0}, (XI.6.53)

and (XI.6.1), we readily prove the existence of an unbounded sequence {r

} ∈

such that (after a possible modiﬁcation of ep by the addition of a constant)

lim

k→∞

∂B



E(x − y)



G(y) · n −

∂ep

∂n

(y)



∂E

∂n

(x − y)ep(y)



dσ

= 0 .

Consequently, (XI.6.52) furnishes the following representation, for all x ∈ Ω

ep(x) =

Ω

G(y) · ∇E(x −y)dy +

∂B

E(x − y)G(y) · n dσ

−

∂B

E(x −y)g(y) dσ

∂B

∂E

∂n

(x − y)ep(y) dσ

(XI.6.54)

where we have used the fact that ∂Ω

= ∂B

. Observing that

∂B

E(x − y) [G(y) · n − g(y)] dσ

= m E(x) +

∂B



E(x − y) − E(x)



[G(y) · n −g(y)] dσ

with

m :=

∂B

[G(y) ·n − g(y)] ,

with the help of (XI.6.53) and the mean-value theorem, from (XI.6.54) we

deduce that

ep(x) = P (x) + m E(x) + O(|x|

−2

) , (XI.6.55)

where

P (x) :=

Ω

G(y) · ∇E(x −y)dy .

We write

P (x) =

Ω

ρ,R/2

G(y) ·∇E(x − y)dy +

Ω

R/2,2R

G(y) · ∇E(x − y)dy

Ω

G(y) · ∇E(x −y)dy

:= P

(x) + P

(x) ,

(XI.6.56)

where |x| = R, suﬃciently large. From (XI.6.5 1) and (XI.6 .1), we readil y

establish, with the help of H¨older’s inequal ity, G ∈ L

(Ω

), and consequently,

employing the properties (XI.6.53) for E, we at once show that

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

(x) + P

(x) = O(|x|

−2

) . (XI. 6.57)

Furthermore, setting B

R,x

:= B

R/2,2R

−B

(x), we have

(x)/R =

R,x

E(x − y)u

(y)u

(y) dy

(x)

E(x −y)u

(y)D

(y) dy

∂B

R,x

E(x − y)u

(y)u

(y)n

dσ

:= P

(x) + P

(x) .

(XI.6.58)

By Theorem XI.6.1, Theorem XI.6.2, and, again, (XI.6.53), we easily show

that

(x)| ≤ c

|x|

−2

1,3R

(x)

|∇∇E(y)|dy ≤ c

|x|

−2

−1

dr ≤ c

|x|

−2

ln |x|,

(x)| ≤ c

|x|

−2

(x)

|∇E(x −y)|dσ

≤ c

|x|

−2

(x)| ≤ c

|x|

−2

∂B

(x)∪∂B

R/2

∪∂B

|∇E(x − y)|dσ

≤ c

|x|

−2

(XI.6.59)

From (XI.6.55)–(XI.6.59) we then deduce

ep(x) = m E(x) + O(|x|

−2

ln |x|) . (XI.6.60)

However, from (XI.6.1), we obtain ep ∈ L

3/2+ε

(Ω

), for all ε positive and close

to zero. Thus, in (XI.6.60 ) we must have m = 0, and the proof of the theorem

is complete. ut

Remark XI.6.1 The establishment of the asymptotic behavior of second-

order derivatives of v, as well as the (related) behavior of ∇p, requires, ap-

parently, a m ore complicated eﬀort, and we leave it unsettled. 

XI.7 On the Asymptotic Structure of Gener alize d

Solutions When v

· ω = 0

Similarly to the irrotational case, the methods we used to investigate the

asymptotic structure of a generalized sol ution corresponding to v

· ω 6= 0,

no longer work if v

· ω = 0. The f unda mental reason resides i n the fact that

we are not able to prove (under suitable a ssumptions on the body force) an

analogue of Theorem XI.4.1, that would ensure that the velocity ﬁeld v + v

∞

XI.7 On the Asymptotic Structure of Generalized Solutions When v

·ω = 0 791

is i n a space L

(Ω

) for some q < 6. Actually, the existence of sol utions that

correspond to v

·ω = 0 and to data of arbitrary “size” and that are in L

a neighbo rhood of inﬁnity for some q ∈ (1, 6) remains an open question.

Notwithstanding this diﬃculty, we can still draw some interesting conclu-

sions on the asympto tic structure of generalized solutions corresponding to

· ω = 0 that, in addi tion, satisﬁes the energy i nequality (XI.2.16). (As we

know from the existence Theorem XI.3.1, this class of solutions is certainly

not empty.) Speciﬁcally, following the work of Galdi & Kyed (2010), we shall

show that provided a certain no rm of the data is suﬃciently small compared to

−1

, every corresponding generalized solution v satisfying the energy i nequal-

ity behaves for large |x| like |x|

−1

. Combining this result with those of Galdi

(2003), one can also deduce ∇v(x), p(x) = O(|x|

−2

), a nd ∇p(x) = O(|x|

−3

However, an important feature that one is able to clarify when v

· ω = 0

and that, as we previously commented, is still obscure in the case v

· ω 6= 0

is the question of the leading terms in the asymptotic expansions of velocity

and pressure ﬁelds, at least when f ≡ v

∗

≡ 0, and T is suﬃciently small.

In particular, as shown by Farwig, Galdi, & Kyed (2011), in such a case the

leading term of the velocity ﬁeld is the velocity ﬁeld of a suitable representative

of Landau solutions, whose class we have recalled in Section X.9; see Theorem

XI.7. 4.

In order to present all the above, we begin by proving the fol lowing

preparatory result.

Lemma XI.7.1 Let Ω be an exterior domain of class C

, and suppose that

the second-order tensor ﬁeld F , and the boundary data v

∗

satisfy the condi-

tions

∇ · F ∈ L

(Ω) , []F []

< ∞,

∗

∈ W

3/2,2

(∂Ω) .

Then, there exists a constant C = C(Ω, q, B) if T ∈ (0, B] such that if

k∇ · F k

+ []F []

+ kv

∗

3/2,2,∂Ω

≤

there is at least one generalized sol ution (v, p) to the Navier–Stokes problem

(XI.0.10)–(XI.0.11) wi th

·ω = 0 and f ≡ ∇ · F such that

u ∈ W

2,2

loc

(Ω) ∩ D

1,2

(Ω) ∩ D

2,2

(Ω) , []u[]

< ∞,

ep ∈ D

1,2

(Ω) ∩ L

(Ω

) for all q

∈ (3/2, 6] and all q

∈ (6, ∞) ,

where u := v + v

∞

, ep is deﬁned in (XI.1.6), and ρ is an arbitrary number

greater than δ(Ω

). Moreover, this solution satisﬁes the estimate

|u|

2,2

+ |u|

1,2

+ []u[]

1,R

+ |ep|

1,2

+ kepk

,Ω

≤ C

(k∇ · F k

+ []F []

+ kv

∗

+ v

∞

3/2,2,∂Ω

) ,

(XI.7.1)

This means that the “body” Ω

, is rotating with “small” angular velocity.

See t he notation in (VIII. 4.46).

In this connection, see Remark VIII.5.1.

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

where C

= C

(Ω, q

, q

, B).

Proof. The proof of this lemma will be obtained by a simple contraction ar-

gument based on the results of Theorem VI II.6.1. Set

X = X(Ω) := {w ∈ D

1,2

(Ω) : []w[]

< ∞}.

Clearly, X becomes a Banach space endowed with the norm kwk

:= |w|

1,2

[]w[]

. Let X

be the ball in X of radius δ centered at the orig in, a nd consider

the map

M : X

→ u = M(w) ∈ X ,

where u solves, for suitable π, the following generalized Oseen problem:

∆u + T (e

×x · ∇u −e

× u) = T w · ∇w + ∇π + f

∇ · u = 0

)

in Ω ,

u = v

∗

+ v

∞

at ∂Ω .

(XI.7.2)

It is clear that if we show that the map M has a ﬁxed point in X

, for some

δ > 0, then the existence of the desired solution (v, p) will be acquired by

setting v := u −e

×x and p := π −

+ x

). If we let F := T w ⊗w + F ,

by assumption and the fact that w ∈ X, we obtain ∇ · F ∈ L

(Ω) with

[]F[]

< ∞. Thus, from Theorem VIII.6.1 there exists a solution (u, π) to

(XI.7.2) such that

u ∈ W

2,2

loc

(Ω) ∩ D

1,2

(Ω) ∩ D

2,2

(Ω) , []u[]

< ∞

π ∈ D

1,2

(Ω) ∩ L

(Ω

) for all q

∈ (3/2, 6], and all q

∈ (6, ∞) ,

(XI.7.3)

where ρ is an arbitrary number greater than δ(Ω

). This shows in particular,

that u ∈ X, so that M is well deﬁned. Furthermore, from (VIII.6.4) in the

same theorem we ﬁnd that u satisﬁes the inequality

|u|

2,2

+ |u|

1,2

+ []u[]

+ |π|

1,2

+ kπk

,Ω

≤ C

(k∇ · Fk

+ []F[]

+ kv

∗

+ v

∞

3/2,2,∂Ω

) ,

where C

= C

(Ω, q

, q

, B) whenever T ∈ (0, B]. Therefore, observing that

k∇ · F k

+ []F[]

≤ T



kwk

∞

|w|

1,2

+ []w[]



+ k∇· F k

+ []F []

≤ 2T kwk

+ k∇ · F k

+ []F []

we infer that in particular, u satisﬁes the following estimate:

kuk

≤ C (T kwk

+ k∇· F k

+ []F []

+ kv

∗

+ v

∞

3/2,2,∂Ω

) . (XI.7.4)

Thus, i f we choose

XI.7 On the Asymptotic Structure of Generalized Solutions When v

·ω = 0 793

δ = 2C (k∇ · F k

+ []F []

+ kv

∗

+ v

∞

3/2,2,∂Ω

)

and require

k∇· F k

+ []F []

+ kv

∗

+ v

∞

3/2,2,∂Ω

≤

from (XI.7.4) it follows that M maps X

into itself. Furthermore, setting

= M (w

), w

∈ X

, i = 1, 2, after a simple calculation tha t uses again

(VIII.6.4), we deduce

−u

≤ C T (kw

+ kw

)kw

− w

. (XI.7.5)

Consequently, since w

∈ X

, i = 1, 2, and δ ≤ 1/(4C T ), from (XI.7.5) it

follows that

− u

≤

− w

which completes the proof that M is a contraction and, as a consequence,

with the help of (XI.7.3), the proof of the lemma.

Remark XI.7.1 Sharper and more detailed asymptotic estimates than those

stated in the previous lemma can be obtained if, in the contraction-mapping

argument used in its proof, we use Theorem VIII.6.2 instead of Theorem

VIII.6.1. More precisely, one can show that under the additional assumption

[|D

+ [|D

< ∞ (XI.7.6)

on F = {F

}, the corresponding generalized solution constructed in Lemma

XI.7. 1 satisﬁes the further asymptotic properties

∇u = O(|x|

−2

) , ep = O(|x|

−2

) , ∇ep = O(|x|

−2

) , as |x| → ∞.

We leave the details of the proof to the interested reader. 

From the previous result in combination with Theorem XI.2.3 we immedi-

ately obtain the foll owing theorem, which furnishes the pointwise asymptotic

behavior of a generalized solution satisfying the energy inequality, a t least for

small data.

Theorem XI.7.1 Let Ω be as in Lemma XI.7.1 and let v be a generalized

solution corresponding to v

· ω = 0 and to da ta

f ∈ L

(Ω) ∩ L

6/5

(Ω), v

∗

∈ W

3/2,2

(Ω) .

Suppose also that f = ∇·F , where F is a second-order tensor ﬁeld in Ω with

[|F |]

< ∞.

Finally, assume that v and the associated pressure p satisfy the

In this connection, see Remark VIII.5.1.

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

energy i nequality (XI.2.16), and that T ∈ (0, B] for some B > 0. Then, there

exists a positive constant c = c(Ω, B) such that if

k∇ · F k

+ []F []

+ kv

∗

3/2,2,∂Ω

≤

, (XI.7.7)

the ﬁelds u = v + v

∞

and ep = p +

+ x

) satisfy all the properties stated

in Lemma XI.7.1. In particular,

v + v

∞

= O(|x|

−1

) as |x| → ∞.

Proof. Let c = min{1/(8C

), 2M }, where C and M are deﬁned in Lemma

XI.7. 1 and Theorem XI.2.3, respectively. Then, in view of (XI.7.7), by Lemma

XI.7. 1 there exists a corresponding generalized solution v

that by (XI.7.1),

satisﬁes in particular the estimate

(|x| + 1)|v

(x) + v

∞

(x)| ≤

Thus, from Theorem XI.2.3 we infer v ≡ v

, and the desired result follows.

Remark XI.7.2 If the tensor ﬁeld F satisﬁes also condition (XI.7.6), then

the generalized solution v of Theorem XI.7.1 and the associated pressure ﬁeld

p satisfy, i n addition the asymptotic properties:

∇(v(x) + v

∞

(x)) = O(|x|

−2

) ,

p +

+ x

) = O(|x|

−2

) , ∇



p +

+ x

)



= O(|x|

−3

) ;

see Remark XI.7.1. 

Theorem XI.7.1 along with Theorem XI. 2.1 and Theorem XI.2.3, furnishes

the following two results whose simple proof is left to the reader.

Theorem XI.7.2 Let the assumptions of Theorem XI.7.1 be satisﬁed. Then

v and the corresponding pressure p satisfy the energy equality (XI.2.1)

Theorem XI.7.3 Let the assumptions of Theorem XI.7.1 be satisﬁed. Then

v is the only generalized solution corresponding to the given data.

As a matter o f fact, the pointwise estimates of Theorem XI.7.1, as well as

those of Remark XI.7.1, can be further reﬁned in the case that f ≡ v

∗

≡ 0

and T is below a certain constant depending on Ω. Actually, as shown by

Farwig, Galdi, & Kyed (2011), in such a case one is able to produce the

leading terms of the asymptotic expansions of v and p. In order to describe

this result, let Ω be of class C

, and denote by v b e the generalized solution

constructed in Theorem XI. 3.1 corresponding to the ab ove data, and by p the

XI.8 Notes for the Chapter 795

associated pressure ﬁeld (see Theorem XI.1.2). Further, l et (U

, P

) be the

Landau solution (X.9.2 1) corresponding to the parameter b = b e

, where

b :=



∂Ω

T (v, ep) · n



· e

, (XI.7.8)

and, we recall, ep is deﬁned in (XI.1.6), while T is the Cauchy stress tensor

(IV.8.6).

The following theorem ho lds.

Theorem XI.7.4 Let Ω be an exterior domai n of class C

and let α ∈ (0, 1).

Then there exists C = C(α, Ω) > 0 such that if T ∈ (0, C], then any gen-

eralized solution (v, ep) to (XI.0.10)–(XI.0.11) correspo nding to v

· ω = 0,

f ≡ v

∗

≡ 0 and satisfying the energy inequality (XI.2.16) satisﬁes the asymp-

totic expansion

v(x) + v

∞

(x) = U

(x) + O



|x|

1+α



as |x| → ∞,

∇



v(x) + v

∞

(x)



= ∇U

(x) + O



|x|

2+α



as | x| → ∞,

(XI.7.9)

and

p(x) +

+ x

) = P

(x) + m ·

4π|x|

+ O



|x|

2+α



as |x| → ∞,

where (U

, P

) is the Landau solution (X.9.21), with b = b e

and b given in

(XI.7.8), while

m := (I −e

⊗ e

) ·

∂Ω



T (v, ep) − (e

×x) ⊗(e

× x)



· n ,

with I the identity matrix.

We shall not give a proof of thi s result, referring the interested reader to the

quoted paper of Farwig, Galdi, & Kyed. We shall limit ourselves to observing

that one of the key points in the proof is the fact that due to its symmetry

properties, the L andau solution o f Theorem XI.7.4 solves (XI.0.10)

1,2

with

R = 0 and f ≡ 0 at each x ∈ R

−{0}. This property was ﬁrst recognized by

Farwig & Hishida (2009).

XI.8 Notes for the Chapter

Section XI.1. A pro of of Corollary XI.1.1 was ﬁrst given by Galdi (2 002,

Theorem 4.6). However, hi s proof is more complicated and less direct than the

one provided here. Furthermore, the assumptions on f are m ore stringent. In