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

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

(x) =

Ω−B

(x)

(y)u

(y)D

(x − y)dy

(x) =

(x)

(x −y)u

(y)D

(y)dy

(x) =

∂B

(x)

(x − y)u

(y)u

(y)n

(y)dy

and s

(2)

satisﬁes (X.5.52)

1,2

. Clearly, in view of (VII.3 .32) and (X.5.5 2)

need to estimate only the term in brackets in (X.8 .28). To this end, we recall

the following bounds (cf. (VII.3.35) and Exercise VII .3.1),

E(x − y)| ≤ c |x − y|

−2

∂B

(x)

E(x − y)| ≤ c R

−1

(X.8.29)

Setting |x| = R and splitting N

as the sum of three integrals, n

, n

, and

, over Ω

R/2

, Ω

R/2,2R

− B

(x), and Ω

, respectively, from (X.8.29)

and

Theorem X.6.4 we deduce

| ≤

Ω

R/2

≤ cR

3/q

−2

kuk

≤ c

|x|

−2+ε

(X.8.30)

where ε is a positive number that can be taken arbitrarily close to zero (at

the cost, of course, of increasing the value of c

). The other two terms, n

and

, are estimated exactly as the i ntegrals I

and I

introduced in the proof of

Theorem X.8.1 (cf. (X.8.22), (X.8.23)) using, this time, (X.8.29)

in place of

(X.8.1)

. We then obtain

| + |n

| ≤ c

|x|

−2+ε

. (X.8.31)

Moreover, bearing in mind that v satisﬁes the uniform estimate (X.8.24), from

(X.8.1)

we recover

|ı

| ≤ c

|x|

−2

. (X.8.32)

It remains to give an upper bound to I

(x). To this purpose, we notice that

from Lemma VII.6.3 it follows f or all suﬃciently large |x|

(x) = −

(x)



(y)D

(d)

(x − y)−u

(y)D

(d)

(x − y)



(X.8.33)

and since, by Theorem X.5.1 ,

|∇u(x)| ≤ M

for suﬃciently large |x| and with M independent of x, we recover

X.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 717

|∇u(x)| ≤ c

|x|

−1

. (X.8.34)

Thus, from (X.8.1)

, (X.8.24), and (X.8.34) we conclude that

(x)| ≤ c

|x|

−2

. (X. 8.35)

Therefore, i dentity (X.8.28) along with (VII.3.32), (X.5.52)

, (X.8.30)–(X.8.32),

and (X.8 .35), allows us to deduce the foll owing theorem.

Theorem X.8.2 Let the assumptions of Theorem X.8.1 hold. Then for all

suﬃciently large |x|, D

v(x), k = 1, 2, 3, admits the following representation:

v(x) = m · D

E(x) + D

(x) (X.8.36)

where E is the Oseen fundamental tensor, m is given in (X.8.18 ) and D

(x)

satisﬁes the estimate

(x) = O(|x|

−2+ε

), (X.8.37)

where the positive number ε can be taken arbitrarily close to zero.

Remark X.8.2 Taking into account the asymptotic properties of D

E(x),

cf. (VII.3.31 ), (VII.3.32), from (X.8.28) it is possible to derive sharper esti-

mates for the g radient of the velocity, according to whether we are or are not

in the paraboloidal wake region. In this regard, it should be observed that the

uniform estimate (X.8.37) can also be slightly improved. For example, Finn

(1959b, Theorem 9) has proved the following bound

(x)| ≤

(

c |x|

−2

, uniformly in x ∈ Ω

c |x|

−2−4σ

, x in the region (X.8.25 ),

(X.8.38)

for a ll σ < σ. For more accurate bounds, we refer the reader to Finn (1965a,

Theorem 5.3) and to Babenko & Vasil’ev (1973, Section 3.3). 

Remark X.8.3 It should be emphasized that the method adopted in the

proof of Theorem X.8.2 do es not apply to higher-order derivatives in the

sense that it does not furnish for such derivatives improved bounds at large

distances. Fo r example, consider D

v(x). We would expect that a s |x| → ∞

v(x) = m ·D

E(x) + O(|x|

−2−α

for som e α > 0. Now, diﬀerentiating twice (X.5.50)

and suitabl y manipulat-

ing the volume integral as we did in the proof of Theorem X.8.2, we obtain

the following relation

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

(x) = m

(x) + D

(2)

−R

Ω−B

(x)

(y)u

(y)D

(x − y)dy

∂B

(x)

(y)D

(x − y)n

(y)dσ

(x)

(y)D

(y))D

(x − y)dy

−R

∂B

(x)

(y)u

(y)D

(x − y)n

(y)dσ

(X.8.39)

Using Lemm a VII.6.3 and the estimate

|∇u(x)| = O(|x|

−3/2

), (X.8.40)

we can show

u(x)| = O(|x|

−3/2

). (X.8.41)

Employing (X.8.40), (X.8.41) into (X.8.39) together with the asymptotic

bounds for E and s

(2)

, and recalling that

v + v

∞

= O(|x|

−1

), (X.8.42)

we can show

(x) = m

(x) + O(|x|

−5/2+ε

)

−R

∂B

(x)

(y)u

(y)D

(x − y)n

(y)dσ

where ε is a positive number arbitrarily close to zero. However, fo r the last

term on the right-hand side of thi s relation we can only say that it behaves as

|x|

−2

for large |x|, and so no improved bound can be deduced on the second

derivatives of v.

The problem of the asymptotic structure of the second derivatives of a

generalized solution has been taken up and sharply solved by Deuring (2005).

In particular, this author proves that, similarly to v and ∇v, also D

behaves “almost” like the corresponding quantities calcula ted for the Oseen

tensor E. More precisely, he shows the following estimate as |x| → ∞

v(x) = α · (D

E(x)) + O



|x|

−5/2

|x|



where α ∈ R

is suitable. In the same paper, an estimate of the asym pto tic

behavior of ∇p is also provided. 

X.8 The Asymptotic Structure of Generalized Solutions when v

∞

6= 0 719

Our next objective is to obtain an asymptotic formula for the pressure

ﬁeld p associated to the generalized solution v. The starting point is the

representation (X.5.50)

, which can be written as

p(x) = p

−M

∗

·e(x) + R

i=1

(x) + h(x) (X.8.43)

with M

∗

and h(x) given by (X.5.51)

and (X.5.52)

, respectively, and

(x) = −R

Ω

R/2

(x − y)u

(y)D

(y)dy

(x) = −R

Ω

R/2,2R

(x − y)u

(y)D

(y)dy

(x) = −R

Ω

(x − y)u

(y)D

(y)dy

where R = |x| . From the inequality

(x − y)| ≤ c|x − y|

−2

(cf. (VII.3 .17)), from (X.8.42) and the uniform bound on the gradient of v

given by (X.8.36), (X.8.38), we easily obtain

(x)| ≤

|x|

kuk

|u|

1,3/2

≤ c

|x|

−2

(x)| ≤

|x|

Ω

R/2,2R

|e(x −y)|dy ≤ c

|x|

−2

(x)| ≤ c

∞

−3

dρ ≤ c

|x|

−2

(X.8.44)

where we have employed the summability properties of u, ∇u as established

in Theorem X. 6.4. Collecting (X.8.43), (X.8.44) and recalling the asymptotic

properties of e and h we then have the following.

Theorem X.8.3 Let the assumptions of Theorem X.8.1 hold and let p be the

pressure ﬁeld a ssociated to v by Lemma X.1. 1. Then there exists a constant p

such that, for all suﬃciently large | x|, p(x) admits the following representation:

p(x) = p

− M

∗

·e(x) + P(x) (X.8.45)

where e is the Oseen fundamental pressure ﬁeld (VII.3. 14), M

∗

is deﬁned in

(X.5.51)

and P(x) satisﬁes the estimate

P(x) = O(| x|

−2

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

Remark X.8.4 Also for the pressure ﬁeld one can give an asymptotic esti-

mate that is sharper than (X.8.45). For example, Finn (1959b, Theorem 10)

has shown that, provided M

∗

is modi ﬁed by adding to it a suitable (constant)

vector, one has

|P(x)| ≤

(

c|x|

−2

uniform ly in x ∈ Ω

c|x|

−2−2σ

x in the region (X.8.25)

for all σ < σ. Further asymptotic estimates on the pressure can be found in

Finn (1965a, Theorem 5.4). 

We end this section by describing the asymptotic structure of the vorticity

ﬁeld ω = ∇×v associated to a solution v to (X.0.8), (X.0.4) with v

∞

6= 0 . This

problem has been studied in full detail by several authors; cf. Clark (19 71),

and Babenko & Vasil’ev (1973, §4). The main result states, essentially, that

if f is of bounded support and v satisﬁes an estima te of the form

v(x) + v

∞

= O(|x|

−1/2−ε

) (X.8. 46)

for some ε > 0, then the principal term in the representation of ω at large

distances is the curl of the principal term of the representation (X. 8.17) for v.

However, as we know from Theorem X.8.1, every generali zed solution obeys

(X.8.46) and so the vorticity ﬁeld of every generalized solution satisﬁes the

preceding property. Thus, in particular, combining Theorem X.8.1 and a the-

orem of Clark (1971, Theorem 3.5) one can show the fol lowing result, whose

rather l ong proof will be omitted.

Theorem X.8.4 Let the assumptions of Theorem X.8.1 be satisﬁed. Then

the vorticity ﬁeld ω = ∇ × v obeys the following representation for all suﬃ-

ciently large |x|

ω(x) = ∇G(x) × m + H(x),

where

G(x) ≡

−ρ

4πR|x|

, ρ =

(|x| + x

m is deﬁned in (X.5.51), and

H(x) = O(e

−ρ

|x|

−2

log |x|).

From this theorem it f ollows, in particular, that the vorticity ﬁeld of any

generalized solution decays exponentially fa st in the region (X.8.25 ) with σ =

1/2, i.e., outside any semi-inﬁnite straight cone of ﬁnite aperture having the

axis coincident with the negative x

-axis.

X.9 On the Asymptotic Structure of Generalized Solutions when v

∞

= 0 721

X.9 On the Asym pt otic St ructure of Generalized

Solutions when v

∞

= 0

The methods we used to investigate the asymptotic structure of a generalized

solution corresponding to v

∞

6= 0, no longer apply when v

∞

= 0. The reason

is ba sically due to the diﬀerent properties possessed at inﬁnity by solutions

to the Oseen and Stokes problems, respectively. More speciﬁcally, what we

cannot do when v

∞

= 0 is to show (under suitable assumptions on the body

force) an analog of Lemma X.6. 1 that would ensure that the velocity ﬁeld v

belongs to L

(Ω

) for some q < 6. As a matter of fact, existence of solutions

corresponding to v

∞

= 0 and to data o f arbitrary “size”, in the class L

in a

neighborhood of inﬁnity, with q ∈ (1, 6), is, to date, an open question.

Nevertheless, by using a completely diﬀerent approach due to Galdi

(1992c), we can stil l draw some interesting conclusion on the asymptotic

structure of generalized solutions corresponding to v

∞

= 0 which, further,

satisfy the energy inequality (X.4.19). (As we know from the existence Theo-

rem X.4.1, this class of solutions is certainly not empty.) Speciﬁcally, we shall

show that, provided a certain norm of the data is suﬃciently small compared

to R

−2

(namely, to the square of the kinematic v iscosity), every corresponding

generalized solution v satisfying the energy inequality behaves for large |x| as

|x|

−1

. Mo reover, employing a simple scaling argument due to

Sver´ak & Tsai

(2000), we can prove that, if f is of bounded support,

the derivatives D

behave like |x|

−|α|−1

, while the derivatives, D

p, of the corresponding pres-

sure ﬁeld p, decay like | x|

−|α|−2

. In other words, v, p possess the asymptotic

properties of the fundamental Stokes tensor U , e. The question of whether

such a result continues to hold for large data also remains open.

It must be also emphasized that, as shown by Deuring & Gal di (2000),

even though v behaves, fo r large |x|, like U , it does not admit an asymptotic

expansion where the leading term is of the form m · U , for some (constant)

non-zero vector m. This issue has been taken up by Nazarov & Pileckas (2000)

and, successively, by Korolev &

Sver´ak (2 007, 2011). In particular, the latter

authors have demonstrated that the leading term coincides with a suitable

exact (and sing ular) solution of the full nonlinear probl em obtained by Landau

(1944); see Remark X.9.3.

The proof of the main result is based on a certain number of steps. To

render the presentation simpler, we shall restrict ourselves to the case where

∗

≡ 0 and shall suppose that the body force f can be written in divergence

form, namely, f = ∇·F , with F a second-order tensor ﬁeld.

For G a vector

or second order tensor ﬁeld, we recall the notation:

[|G|]

≡ sup

x∈Ω



(1 + |x|

)|G(x)|



Concerning this assumption, see footnote 4.

This latter condition is not, in fact, a restriction, provided we give some regularity

on f ; cf. Exercise III.3.1. See also Lemma VIII.5.1.

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

We begin with the following.

Lemma X.9.1 Let Ω ⊆ R

be an exterior domain of class C

. Suppose that

the second order tensor ﬁeld F in Ω satisﬁes

(1 + |x|

)F ∈ L

∞

(Ω).

Then there exi sts a positive constant A = A(Ω, q) such that if

[|F |]

< A/R

(X.9.1)

there is at l east one generalized solution v to the Navier–Stokes problem

(X.0.8), (X.0.4) with v

∗

≡ v

∞

≡ 0 and f = ∇ · F

such that

v ∈ D

1,q

(Ω), p ∈ L

(Ω) for each q > 3/2 ,

[|v|]

< ∞.

Moreover, for any q > 3/2, this solution satisﬁes the following estimate

[|v|]

+ |v|

1,q

+ kpk

≤ BR[|F |]

, (X.9.2)

with B = B(Ω, q).

Proof. The solution can be determined (for instance) by the successive approx-

imation method. Thus, we look for a sequence of sol utions to the sequence of

problems, m ∈ N,

(∇v

, ∇ψ) + (v

m−1

· ∇v

m−1

, ψ) − (p

, ∇· ψ) − (F , ∇ψ) = 0

∈ D

1,q

(Ω), p

∈ L

(Ω)

(X.9.3)

where v

≡ 0 and ψ is arbitrary from C

∞

(Ω). By the existence theory for the

Stokes problem of Theorem V.8.1, we know that (X.9.3) with m = 1 admits

a unique solution {v

, p

} such that for all q > 3/2

[|v

+ |v

1,q

+ kpk

≤ 2cR[|F |]

(X.9.4)

where c = c(q, Ω) is the constant entering the estimate (V.8.25 ). Let us show,

by induction, the existence of {v

, p

} satisfying (X.9.4) for all n ∈ N. Thus,

assume {v

m−1

, p

m−1

} obeys (X.9.4). Since

m−1

· ∇v

m−1

= ∇ · (v

m−1

⊗ v

m−1

)

[|v

m−1

≤ [|v

m−1

< ∞,

from Theorem V.8.1 we establish the existence of v

, p

obeying (X.9.3)

along with the estimate

In the weak sense.

X.9 On the Asymptotic Structure of Generalized Solutions when v

∞

= 0 723

[|v

+ |v

1,q

+ kp

≤ cR



[|F |]

+ [|v

m−1



. (X.9.5)

However, by the induction hypothesis,

[|v

m−1

≤ 4c

[|F |]

and from (X.9.5) we ﬁnd

[|v

+ |v

1,q

+ kp

≤ cR[|F |]



1 + 4 c

[|F |]



. (X.9.6)

Therefore, i f (X.9.1) holds with A = 1/4c

, the approximating solution satis-

ﬁes, for all m ∈ N, the following inequality

[|v

+ |v

1,q

+ kp

≤ 2cR[|F |]

. (X.9.7)

Notice that this inequality coincides with (X.9.2) with v

in place of v and

B = 2c. It is now a standard procedure, which we already employed several

times, and which we therefore omit here (see, for instance, the proof of Theo-

rem IX.5.3), to show that if (X.9.7) is veriﬁed then v

, p

converges strongly

in D

1,q

(Ω) × L

(Ω) to functions v, p such that

v ∈ D

1,q

(Ω), p ∈ L

(Ω).

Moreover,

[|v|]

< ∞

and (X.9 .2) is satisﬁed. The theorem is thus proved. ut

Remark X.9.1 Lemma X.9.1, under the same a ssumptions, remains valid in

any number of dimensions n ≥ 4, the only change in its statement being that

the restriction from below on q b ecomes q > n/2. 

Exercise X.9.1 (Finn (1965a)) Show t hat, for suﬃciently small R, the solution

constructed in Lemma X.9.1 can be expanded as a p ower in R:

v(x) = v

(x) +

∞

k=1

(x), x ∈ Ω, (X.9.8)

where v

(x) is a solution to the following Stokes problem

∆v

= ∇p

+ RF

∇·v

= 0

)

in Ω

= 0 at ∂Ω

lim

|x|→∞

(x) = 0.

Hint: A solution of the form (X.9.8) can be constructed by recurrence, that i s,

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

∆v

m+1

j=0

m−j

· ∇v

+ ∇p

m+1

∇· v

m+1

= 0

)

in Ω

m+1

= 0 at ∂Ω

lim

|x|→∞

m+1

(x) = 0

m = 0, 1, 2, . . . . This sequence of problems can then be solved with the help of

Theorem V.8.1. Moreover,

m+1

≤ c

j=0

m−j

(X.9.9)

where c = c(Ω) and V

= [| v

+ |v

1,q

. By (X.9.9) and the Cauchy product

formula for the series, the series

∞

k=0

is converging whenever R < 1/4cV

, that is, since V

≤ cR[|F |]

, whenever R satisﬁes

a restriction of the type (X.9.1). Finally, the uniqueness Theorem X.3.2 implies that

(X.9.8) coincides with the solution constructed in Lemma X.9.1.

Lemma X.9.2 Let v be a generali zed solution to the Navier–Stokes problem

(X.0.8), (X.0.4) corresponding to v

∗

≡ v

∞

≡ 0 and f of bounded suppo rt. If

v(x) = O(|x|

−1

), as |x| → ∞, then

v(x) = O(|x|

−|α|−1

) ,

(p(x) −p

) = O(|x|

−|α|−2

)

for som e p

∈ R and for all |α| ∈ N .

Proof. To show the decay of the velocity ﬁeld, we follow the scaling argument

Sver´ak & Tsai (2000).

Fix x ∈ Ω, set R = |x|/3 suﬃciently large so that

supp (f ) ⊂ Ω

, and deﬁne

(y) := R v(Ry + x) , p

(y) := R

p(Ry + x) , (X.9.10)

where p is the pressure ﬁeld associated to v by Lemma IX.1.2. Since, by

Theorem X.1.1, v, p ∈ C

∞

(Ω

), and satisfy in Ω

(X.0.8)

1,2

with f ≡ 0, it

easily follows that v

, p

is a smooth solution to the following system

∆v

= Rv

·∇v

+ ∇p

∇ · v

= 0

)

in B

. (X.9.11)

We now apply to (X.9.11) the interior estimate for the Stokes problem given

in (IV.4.20) to obtain, in particular,

Actually, the result of

Sver´ak & Tsai only requires that f decays suﬃciently

rapidly (depending on |α|) as |x| → ∞.

X.9 On the Asymptotic Structure of Generalized Solutions when v

∞

= 0 725

1,s,B

≤ c



s,B

+ kv

2s,B



. (X.9.12)

In view of the assumption on v, we may take s is arbitrary in (1, ∞), and thus

obtain

1,s,B

≤ M , (X.9.13)

for all s ∈ (1, ∞) for a constant M independent of R. We then use (IV.4.4),

Remark IV.4.1 along with the embedding Theorem II. 3.4 to ﬁnd

2,s,B

≤ c



1,s,B

+ kv

1,s,B



where a < 1. This inequality combined with (X.9.13) yields

2,s,B

≤ M

(X.9.14)

with M

independent of R, and then by the embedding Theorem II.3.4,

max

y∈B

|∇v

(y)| ≤ M

with M

independent of R. Therefore, from (X.9 .10) it follows that

|∇v(x)| ≤ M

which shows the result for k = 1. The estimate of the higher order deriva-

tives is obtained by a simple boot-strap argument. Thus, f or example, f rom

(IV.4.4), Remark IV.4.1 , the embedding Theorem II.3.4, and (X.9.1 4) we de-

duce kv

3,s,B

≤ M

, b < a, which, by (X.9.10 ), and Theorem II.3.4 in turn

implies R

v(x)| ≤ M

, and so forth. We next come to the estimate for the

pressure. For |α| ≥ 1 they immediately follow from the estimate just proved

for v and from (X.0.8)

(with f ≡ 0). In order to show the stated decay for

|α| = 0, we notice that from (X.5.47)

, (VII.3. 14) and (X.5.47)

we ﬁnd, for

some p

∈ R, and all suﬃciently large |x| := 3R

p(x) = p

− R

Ω

(x − y)v

(y)D

(y)dy + O(1/|x|

)

= −R

Ω

R,2R

(x − y)v

(y)D

(y)dy − R

Ω

(x − y)v

(y)D

(y)dy

+O(1/|x|

)

:= RI

(R) + RI

(R) + O(1/|x|

) .

(X.9.15)

Recalling that |q(ξ)| ≤ c |ξ|

−2

, the a symptotic estimates for v, and that |x| =

3R we at once obtain

(R)| ≤

Ω

R,2R

|v · ∇ v| ≤

|Ω

R,2R

| ≤

|x|

(R)| ≤

|x|

|x −y|

|y|

≤

|x|

(X.9.16)