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

Подождите немного. Документ загружается.

506 VIII Steady Generalized Oseen Flow in Exterior Domains

−∇ × (ψ

∇× v), and integrate over Ω. Since

−∇ × (ψ

∇ × v) = −ψ

∇× ∇ × v + (∇ × v) ×∇ψ

= ψ

∆v + (∇ × v) × ∇ψ

(VIII.2.4)

we obtain

√

∆vk

= −



∂v

∂x

+ f, ψ

∆v + (∇ × v) × ∇ψ



−

((∇ × v) × ∇(

√

∆v)

+T (e

× v, ∇× (ψ

∇ × v) − (e

× x · ∇v, ∇× (ψ

∇ × v)) .

(VIII.2.5)

By the Schwarz i nequality, the Cauchy inequali ty (II.2.5), and the properties

of ψ

, we obtain

− ((∇ × v) × ∇(

∆v) ≤

∆vk

+ M

|v|

1,2

(VIII.2.6)

and

−



∂v

∂x

+f , ψ

∆v+(∇ × v) × ∇ψ



≤

∆vk

+ c (|v|

1,2

+ kfk

) ,

(VIII.2.7)

where c = c(B, ρ, r). We need now to estimate the last two terms on the

right-hand side of (VIII.2.5). Employing the well-known identities

∇ · (A ×B) = B · ∇ ×A − A · ∇ × B ,

∇ × (a × B) = a ∇ · B − a ·∇B ,

(VIII.2.8)

where A, B are vector ﬁelds and a is a constant vector, along with the prop-

erties of ψ

, we obtain

× v, ∇ × (ψ

∇ × v)) = −

Ω

∇ · [ψ

×v) × (∇ × v)]

+ (ψ

∇× v, ∇ × (e

×v))

= −(ψ

· ∇v, ∇×v) ≤ 2|v|

1,2

(VIII.2.9)

Furthermore, a gain with the help of (VIII.2.8)

, and by the properties of ψ

we deduce

This vector identity as well as the others used throughout the proof is known for

“smooth” vector ﬁelds. However, by a simple approximating procedure based on

Theorem II.3.1, we can show that they continue to hold a.e. in Ω

also if the

ﬁelds merely belong to W

2,2

loc

(Ω

Throughout t he proof, for simplicity, we set k ·k

2,Ω

≡ k·k

, and (·, ·)

Ω

≡ (·, ·).

VIII.2 Generalized Solutions. Uniqueness 507

−(e

× x · ∇v, ∇× (ψ

∇ × v)) =

Ω

∇ · [ψ

× x · ∇v) × (∇×v)]

−(ψ

∇ × v, ∇× (e

×x ·∇v))

= −(ψ

∇ × v, ∇× (e

×x ·∇v)) .

(VIII.2.10)

In order to estimate the term on the right-hand side of (VIII.2.10), we recall

the following classical form o f ∇ × A, and that of the cross product of two

vectors, a, b in terms of their components:

∇× A = ε

ijk

, a × b = ε

ijk

where ε

ijk

is the alternating symbol.

We thus have, for i = 1, 2, 3,

[∇× (e

×x · ∇v)]

= ε

ijk

× x · ∇v

)

= [e

×x ·∇(∇ × v)]

+ ε

ijk

lmj

and so, employing the well- known identities (see, e.g., Evett (1966))

ijk

lmj

= δ

− δ

and recalling (VIII.2.3)

, we deduce

∇ × (e

× x · ∇v) = e

× x · ∇(∇ × v) − ∇v · e

Taking into account that e

×x ·∇ψ

(|x|) = 0, for all x ∈ R

, from this latter

relation and again from the properties of ψ

, we obtain

−(ψ

∇ × v, ∇×(e

× x · ∇v)) = −

Ω

∇ · [ψ

(∇ × v)

×x]

+ (ψ

∇ × v, ∇v · e

)

= (ψ

∇ × v, ∇ v ·e

) ≤ 2|v|

1,2

(VIII.2.11)

Collecting (VIII.2.5)–(VIII.2.7), (VII I.2.9), and (VIII.2.11), we thus obtain

∆vk

≤ c (|v|

1,2

+ kfk

) , (VIII.2.12 )

and so, recalling that ψ

(|x|) = 1, for x ∈ Ω

r,R

, we conclude, in particular,

that

k∆vk

2,Ω

r,R

≤ c (| v|

1,2

+ kfk

)

where c is independent of R. Letting R → ∞ in this relation gives ∆v ∈

(Ω

), for all r > ρ, and moreover,

k∆vk

2,Ω

≤ c (|v|

1,2

+ kfk

2,Ω

) . (VIII.2.13)

See footnote 4 in Section VIII.1.

508 VIII Steady Generalized Oseen Flow in Exterior Domains

From (VIII.2.13) and from its proof, it immediately follows that in the par-

ticular case Ω = R

and f ∈ L

), we have ∆v ∈ L

) and

k∆vk

2,R

3 ≤ c

(|v|

1,2

+ kfk

2,R

3 ) , (VI II.2.14)

with c

= c

(B). We now observe that as a consequence of this property and

the Helmholtz–Weyl decomposition theorem, Theorem III.1.1 , v satisﬁes the

following Stokes system:

∆v = F + ∇ζ

∇ · v = 0

)

in Ω

where F ∈ H(Ω

) ⊂ L

(Ω

) and ζ ∈ D

1,2

(Ω

). Moreover, v ∈ D

1,2

(Ω

) by

assumption, and so f rom Theorem V.5.3, we readily obtain v ∈ D

2,2

(Ω

), and

the property stated in the lemma follows. Clearly, if Ω = R

and f ∈ L

this argument furnishes D

v ∈ L

). Therefore, since

v ∈ D

2,2

) ∩ D

1,2

) ,

by Theorem II.7.6 it easily follows that v ∈ D

2,2

). Thus, by Exercise II.7.4,

we have

= k∆vk

and from (VIII.2.14) we also prove (VIII.2.2). The case Ω 6= R

can be treated

by similar considerations. In fact, l et ζ = ζ(x) a smooth function that is 1

for | x| ≥ r and is 0 for |x| ≤ ρ, and set w = ζ v. Then, by the property

of generalized solutions and with the help of Theorem II.7.6 we prove that

w ∈ D

2,2

), and so, as before, we ﬁnd that kD

= k∆wk

. However,

by a simple calcula tion, we show that

2,Ω

≤ k∆wk

≤ c

(r) (k∆vk

2,Ω

+ kvk

1,2,Ωρ,r

)

≤ c

(r) (k∆vk

2,Ω

+ |v|

1,2

) ,

(VIII.2.15)

where in the last step, we have used the H¨o lder inequality together with

inequality kvk

≤ c| v|

1,2

. The relation (VIII.2.1) is then a consequence of

(VIII.2.13) and (VIII.2.15).

The next result furnishes a representation of the pressure ﬁeld under very

general assumptions on f .

Lemma VIII.2.2 Let v be a generalized soluti on to (VIII.0.2), (VIII.0.7)

corresponding to f =

f + ∇ · F , where

f ∈ L

(Ω

) ∩ L

(Ω

), for some ρ > δ(Ω

) and q ∈ (1, ∞) ,

and F is a second-order tensor ﬁeld such that

VIII.2 Generalized Solutions. Uniqueness 509

∇· F ∈ L

(Ω

) , F ∈ L

(Ω

) ∩ L

(Ω

) ∩ L

(Ω

) , for some t ∈ (1, ∞).

Then, there exists p

∞

∈ R such that the pressure p associated to v by L emma

VIII.1.1 admits the following decomposition:

p = p

∞

+ p

, (VIII.2.16 )

where

∈ L

(Ω

) ∩D

1,2

(Ω

) ∩ D

1,q

(Ω

) ,

∈ L

(Ω

) ∩L

(Ω

) ∩ D

1,2

(Ω

) .

(VIII.2.17)

Moreover, if q ∈ (1, 3), we have al so

∈ ∩L

3q/(3−q)

(Ω

) . (VIII. 2.18)

Finally, if

(f, ∇ ψ) = 0 for all ψ ∈ C

∞

(Ω

) , (VIII.2.19)

namely, ∇ ·f = 0 in the generalized sense, and q ∈ (1, 3/2], t ∈ (1, 3], then

(p(x) − p

∞

) = χ

(x) , for all |α| ≥ 0, |x| > r , (VIII.2.20)

where

(x) = O(|x|

−2−|α|

) .

Proof. By the Helmholtz–Weyl decomposition theorem, Theorem III.1.2, we

may write

f = f

∗

+ ∇p

∗

where f

∗

, ∇p

∗

∈ L

(Ω

) ∩L

(Ω

). By Theorem II.6.1, we can add a constant

to p

∗

in such a way that the modiﬁed function, which we continue to denote

by p

∗

, belongs to L

(Ω

)∩L

3q/(3−q)

(Ω

). We set ep = p+p

∗

, so that (VIII.2.3)

becomes

∆v + R

∂v

∂x

+ T (e

× x · ∇v − e

× v) = ∇ep + f

∗

+ ∇· F

∇· v = 0











a.e. in Ω

(VIII.2.21)

We begin by proving the properties (VIII.2.16)–(VIII.2.18). Fix r > ρ, and

let ψ be a smooth “cut-oﬀ” f unction that is 1 if |x| ≥ r, and 0 if |x| ≤ ρ.

Moreover, let

σ(x) := −



∂B

v · n



∇E , (VIII.2.22)

where E is the (three-dimensional) fundamental Laplace solution (II.9.1), and

consider the following problem:

510 VIII Steady Generalized Oseen Flow in Exterior Domains

∇ · H = ∇ · [ψ(v + σ)] in B

H ∈ W

3,2

) , supp (H) ⊂ B

(VIII.2.23)

Since

∇ · [ψ(v + σ)] =

∂B

ψ(v + σ) · n = 0 ,

we know, by the properties of v and Theorem II I.3.1, that (VIII.2.23) has at

least one solution. Put

w = ψv + ψσ −H , π = ψ ep.

Taking into account (VIII.1.8), from (VIII. 2.21) by a direct calculation we

obtain

∆w +

∂w

∂x

+ (e

× x · ∇w − e

×w) = ∇π + ψf

∗

+ ψ∇·F + G

∇ · w = 0











a.e. in R

(VIII.2.24)

where G ∈ L

), with supp (G) ⊂ B

. Now, for all ψ ∈ C

∞

(Ω

(∆w, ∇ψ) = −(∇ · w, ∆ψ) = 0 ,



∂w

∂x

, ∇ψ





∇ · w,

∂ψ

∂x



= 0 .

(VIII.2.25)

Also, since

∇ · (e

×x ·∇Φ − e

× Φ) = e

× x · ∇(∇ · Φ) = 0 ,

for all Φ ∈ W

2,2

loc

(Ω) with ∇ ·Φ = 0 ,

(VIII.2.26)

we obtain, in particular,

×x·∇w−e

×w, ∇ψ) = −(∇·(e

×x·∇ w−e

×w), ψ) = 0 . (VIII.2.27 )

Using (VIII.2.25) and (VIII.2.27) from (VIII.2.24)

, we deduce that π is a

generalized solution to the foll owing problem:

− ∆π = ∇ ·(ψf

∗

) + ∇ · (ψ∇ · F ) + ∇ · G in R

. (VIII.2.28)

A solution to (VIII. 2.28) is given by

π = −∇E ∗ (ψf

∗

) −∇E ∗ G + ∇E ∗ (∇ψ · F ) − ∇E ∗ [∇· (ψF )]

≡

i=1

(VIII.2.29)

Since their values are irrelevant to the proof, we set throughout R = T = 1.

VIII.2 Generalized Solutions. Uniqueness 511

where E i s the fundamental solution of Laplace’s equation (II.9.1). We recall

that |∇E(ξ)| ≤ c|ξ|

−2

, and that D

E is a singular kernel, for all i, j = 1, 2, 3

(see Exercise II.11.7). Therefore, ta king into account that ψf

∗

, G, ∇ψ · F ∈

) ∩ L

), q ∈ (1, 3), from Theorem II.11.3 and Theorem II. 11.4 we

obtain

∈ L

) ∩ D

1,q

) ∩ D

1,2

), q ∈ (1, ∞)

∈ L

3q/(3−q)

) ∩ L

) ∩ D

1,q

) ∩ D

1,2

), q ∈ (1, 3)

)

i = 1, 2, 3,

(VIII.2.30)

and by the same token,

∈ L

) ∩ L

) ∩ D

1,2

) ; (VIII.2.31)

see Exercise VIII.2.2 . L et us show that (up to an additive constant) π = π in

Ω

. To this end, we observe that clearly, Ψ := π − π is harmonic in R

, for

which the fol lowing local representation holds (see (VI I.4. 57)):

Ψ(x) = −

(R)

(x − y)Ψ(y)dy x ∈ R

with H

(R)

satisfying (V. 3.11)–(V.3.13). Therefore, by diﬀerentiating bo th

sides of this relation and using the Schwarz and H¨older inequalities, we deduce

|∇Ψ (x)| ≤



|y|

(R)

(x − y)|



1/2



∇π



+k∇H

(R)

3q/(4q−3)

kπk

3q/(3q−3)

(VIII.2.32)

Also, recalling the properties (V. 3.11), (V.3.13) of the function H

(R)

and

noticing that |y|

≤ 2(|x − y|

+ |x|

), from (VIII.2 .32) we obtain

|∇Ψ (x)| ≤ c

1 + | x|

√



∇π



+ c

−5+4q/3

kπk

3q/(3q−3)

, (VIII.2.33)

for a ll ﬁxed x ∈ R

and all R > 0 . On the other ha nd, dividing b oth sides of

(VIII.2.21)

by |x|, squaring the resulting equation and integrating over Ω

r > ρ, we deduce



∇ep

|x|



2,Ω

≤ c

k∆vk

2,Ω

+ |v|

1,2,Ω

+ kfk

2,Ω



|x|



2,Ω

, (VIII.2.34)

where c = c(r). From Lemma VIII.2.1 and the assumption on f , we infer that

∆v ∈ L

(Ω

). Therefore, taking into account that v ∈ D

1,2

(Ω), we deduce

that the ﬁrst three terms on the right-hand side of (VIII.2.34) are ﬁnite.

Moreover, by Theorem II.6.1(i) and property (iv) in Deﬁnition VIII.1.1, we

have also that the last term on the right-hand side of (VIII.2.34) is ﬁnite, so

that we conclude, in particular, that

512 VIII Steady Generalized Oseen Flow in Exterior Domains

Ω



∇ep



< ∞,

namely, since π = ψep and ep ∈ L

loc

(Ω



∇π



< ∞.

If we take into account this conditio n al ong with (VIII.2.30), and pass to the

limit R → ∞ in (VIII.2 .33), we conclude that ∇Ψ(x) = 0, for all x ∈ R

that is, π = π + const. Therefore, since π(x) = ep(x) ≡ p(x) + p

∗

(x), x ∈

Ω

, the desired summability property (VIII.2.16)–(VIII.2.17) for p fol lows

from thi s latter, the analogous property of p

∗

, and (VIII.2.29)–(VIII.2.31). We

shall now complete the pro of of the theorem. Using (VIII.2 .25), (VIII.2.27)

(with w ≡ v), and bearing in mind (VIII.2.19), from (VIII.2.2 1) we obtain

(∇p, ∇ψ) = 0, for a ll ψ ∈ C

∞

(Ω

), whi ch, i n turn, by well-known results

of Ca ccioppoli (1937), Cimmino (1 938a, 19 38b), and Weyl (1940), i mplies

that p is harmonic in Ω

and belongs to C

∞

(Ω

). Recalling that p satisﬁes

(VIII.2.16)–(VIII.2.17), we may use the results of Exercise V.3.6(i) to obtain

the following representation for ep(x), for al l x ∈ Ω

p(x) = −



∂B

∂ep

∂n



E(x) + χ

(x) , for all |α| ≥ 0. (VIII.2.35)

Take α = 0 in this relation. Then, if q ∈ (1, 3/2], t ∈ (1, 3], by (VIII. 2.16),

(VIII.2.17) both p and χ

are summable in Ω

with some exponent s ∈

(3/2, 3], and since E 6∈ L

(Ω

), this impl ies that the surface integral in

(VIII.2.35) must vanish, which concludes the proof of the lemma. ut

Remark VI II.2.1 As the reader may have noticed, the proof of the previous

lemma remains una ltered if we take T = 0. 

We are now in a position to prove the following.

Lemma VIII.2.3 Let Ω be locally Lipschitz and let u be a generalized so-

lution to (VI II.0.2), (VIII.0.7) correspondi ng to f ≡ v

∗

≡ 0. Moreover, de-

note by p the associated pressure ﬁeld from Lemma VIII.1.1. Then u ≡ 0,

p ≡ p

+ const.

Proof. By assumption, we have

(∇u, ∇ψ)−R



∂u

∂x

, ψ



−T (e

×x·∇u−e

×u, ψ) = (π, ∇·ψ) , (VI II.2.36)

for all ψ ∈ C

∞

(Ω). By Theorem I II.5.1, we obtain that u ∈ D

1,2

(Ω), whi le

by Theorem VIII.1.1, it also follows that u, p ∈ C

∞

(Ω). Employing this latter

property and (VIII.2.36), we deduce that

VIII.2 Generalized Solutions. Uniqueness 513

∆u + R

∂u

∂x

+ T (e

×x · ∇u −e

×u) = ∇π

∇ · u = 0











in Ω . (VIII.2.37)

Now by Lemma VI II.2.2, we have

π(x) − π

∞

= O(|x|

−2

) , as |x| → ∞, (VIII.2.38)

for some π

∞

∈ R. Since (VIII.2.36) remains unchanged if we replace π with

π −π

∞

, we will take, without loss, π

∞

= 0. Our next objective i s, basically, to

replace ψ in (VIII.2.36) with u. In order to reach this goal, we begin to notice

that since D

1,2

(Ω) ⊂ W

1,2

(Ω

), for all R > δ(Ω

), one can show, by a simple

density argument, that (VIII.2.36) continues to hold for ψ ∈ W

1,2

(Ω

), for

all R > δ(Ω

). We next choose ψ = ψ

4,R

u, where ψ

α,R

is the “cut-oﬀ”

function determined in Lemma II.6.4. We recall that by that lemma, we have,

in particular,

supp (ψ

4,R

) ⊂ Ω

√

lim

R→∞

4,R

(x) = 1 uni formly pointwise ,



∂ψ

4,R

∂x



3/2

≤ C

, ku |∇ψ

4,R

≤ C

|u|

1,2,Ω

√

× x) · ∇ψ

4,R

(x) = 0 for all x ∈ R

(VIII.2.39)

Thus, replacing ψ

4,R

u for ψ in (VIII.2.36), we obtain after a few integrations

by parts (for notati onal simplicity, we drop the subscript “

”),

0 = (ψ

∇u, ∇u) + (∇ψ

· ∇u, u) +



∂ψ

∂x



×x · ∇ψ

, u

) − (π, ∇ψ

· u)

= (ψ

∇u, ∇u) + (∇ψ

· ∇u, u) +



∂ψ

∂x



−(π, ∇ψ

· u) .

From this relation, with the help of (VIII. 2.39) and the Schwarz and H¨older

inequalities, we derive

∇uk

≤ c



|u|

1,2,Ω

√

+ Rkuk

6,Ω

√

+ kπk

|u|

1,2,Ω

√



, (VIII.2.40)

with c independent of R. From (VIII.2.38 ) we k now that π ∈ L

(Ω), whereas

by Theorem II.7.5, we have u ∈ L

(Ω). Therefore, in v iew of these properties,

we may let R → ∞ in (VIII.2 .40) to deduce

lim

R→∞

∇uk

= 0 ,

514 VIII Steady Generalized Oseen Flow in Exterior Domains

which, in turn, by (VIII.2.39)

and by the Lebesgue dominated convergence

theorem of Lemma II.2.1, shows that k∇uk

= 0, that is, u ≡ 0 in Ω. The

proof of the theorem is thus completed. ut

From the previous lemma, we at once deduce the following uniqueness

theorem, which constitutes the main accomplishment of this section.

Theorem VIII.2.1 Let Ω be locally Lipschitz and let v be a generalized

solution to (VIII.0.2), (VIII.0 .7) corresponding to f ∈ W

−1,2

(Ω

), Ω

any

bounded subdomain with Ω

⊂ Ω, and to v

∗

∈ W

1/2,2

(∂Ω). Moreover, denote

by p the asso ciated pressure ﬁeld from Lemma VIII.1.1. Then, if w is another

generalized solution corresponding to the same data, with associated pressure

ﬁeld p

, we have v ≡ w, p ≡ p

+ const.

Exercise VIII.2.1 Let v be a generalized soluti on to (VIII.0.2), (VIII.0.7), and

suppose that f satisﬁes the assumption of Lemma VIII .2.1. Show that

lim

|x|→∞

v(x) = 0 ,

uniformly pointwise. Hint: Couple the results of Lemma VIII.2.1 with those of The-

orem II.9.1.

Exercise VIII.2.2 Prove properties (VIII. 2.30), (VIII.2.31).

VIII.3 The Fundamental Solution to the

Time-Dependent Oseen Problem and Related Properti es

As we observed in the Introduction to thi s chapter, the asymptotic proper-

ties o f generalized solutions to (VIII.0.7), (VII I.0.2) will be established by

ﬁrst proving similar properties for the solutions to an associated i nitial-value

problem, and then by showing that in the limit as t → ∞, the latter converge

to the former, uniformly in suitable no rms.

Actually, as we shall show later on, in order to achieve this goal, it suﬃces

to investigate the above properties for solutions to the following Oseen initial-

value problem:

∂u

∂t

= ∆u + R

∂u

∂x

−∇φ + f

∇ · u = 0











in R

× (0, ∞)

u(x, 0) = u

(VIII.3.1)

where f = f (x, t) and u

= u

(x) are given vector ﬁelds, with u

solenoidal,

satisfying appropriate assumptions. In turn, the study of the asymptotic prop-

erties (in space and time) of solutions to probl em (VIII.3.1) is more conve-

niently done by means of the integral representation of the solutions through

VIII.3 Fundamental Solution to the Time-Dependent Oseen Problem 515

the fundamental tensor solution o f the Oseen system (VIII.3.1)

1,2

. Such a

solution is well known, and was originally introduced by Oseen (1927, §5).

The objective of this section will thus be to recall some of the relevant

properties of the Oseen fundamental solution, and to establish certain uniform

properties concerning its asymptotic behavior i n space and ti me.

Since the study of all principal properties of this solution will be performed

in great depth in Volume II,

we shall restrict ourselves to state those we need

here, without detailed proofs, referring the reader to the existing literature for

all the missing details.

Following Oseen (1927, §5), for all R ≥ 0 we introduce tensor and vector

ﬁelds, Γ and γ, respectively, deﬁned by

(x −y, t −τ; R) = −δ

∆Ψ(|x + R(t − τ )e

− y|, t − τ )

∂

∂y

Ψ(|x + R(t − τ )e

−y|, t − τ )

(x − y, t −τ) =

∂

∂y

(∆ +

∂

∂τ

)Ψ(|x + R(t − τ )e

−y|, t − τ ) ,

(VIII.3.2)

where Ψ = Ψ (r, s) is any real function that is deﬁned and smooth for all

r ≥ 0 and all s > 0. By a straightforward calculation, we show that the ﬁelds

(VIII.3.2) satisfy the following equations:

∂Γ

∂τ

− R

∂Γ

∂y

+ ∆Γ

−

∂γ

∂y

= −δ



∂

∂τ

+ ∆



∆Ψ ,

∂Γ

∂y

= 0 ,

(VIII.3.3)

where “∆” operates on the y-variables. The idea is now to choose Ψ in such

a way that

∆Ψ(r, s) = −W (r, s) , (VII I.3.4)

where W = W (r, s) is the three-dimensional Weierstrass kernel, namely, the

fundamental solution to the three-dimensional heat equation:

W =

(

(4π s)

−3/2

−

, s > 0 ,

0 , s ≤ 0 .

(VIII.3.5)

This choice will then ensure that the right-hand side of (VIII.3.3)

vanishes

for x, y ∈ R

and t > τ, and moreover, that when τ → t

−

, it becomes

“appropriately singular,” in a way that will be clariﬁed later on, in Lemma

VIII.3.1. The requirement (VIII.3.4) means that

Actually, in any space dimensions n ≥ 2.

The properties that we will state in this section continue to hold also for R < 0.

However, for the application we have in mind, we suppose that R is nonnegative.