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

Подождите немного. Документ загружается.

446 VII Steady Oseen Fl ow in Exterior Domains

Rather, we shall make use of a more appropriate tool due to P.I. Lizorkin

(1963, 1 967), which we are going to describe.

Denote by S(R

) the space of functions of rapid decrease consisting of

elements u from C

∞

) such that

sup

x∈R



· . . . · |x

u(x)|



< ∞

for a ll α

, . . ., α

≥ 0 a nd |β| ≥ 0. For u ∈ S(R

) we denote by bu its Fourier

transform:

bu(ξ) =

(2π)

n/2

−ix·ξ

u(x)dx,

where i stands for the imaginary unit. It is well known that bu ∈ S(R

) and

that, moreover,

u(x) =

(2π)

n/2

ix·ξ

bu(ξ)dξ,

see, e.g., Reed & Simon (1975, Lemma on p. 2). Given a function Φ : R

→ R,

let us consider the i ntegral transform

T u ≡ h(x) =

(2π)

n/2

ix·ξ

Φ(ξ)bu(ξ)dξ, u ∈ S(R

). (VII.4.4)

Generalizing the works of Marcinkiewicz (1939) and Mikhlin (1957), Li-

zorkin (1963, 1967) has proved the following result, w hich we state witho ut

proof.

Lemma VII.4.1 Let Φ : R

→ R be continuous together with the derivative

∂

∂ξ

. . .∂ξ

and al l preceding derivatives for |ξ

| > 0, i = 1, . . ., n. Then, if for some

β ∈ [0, 1) and M > 0

|ξ

+β

·. . . ·|ξ

+β



∂

∂ξ

. . .∂ξ



≤ M,

where κ

is zero o r one and κ =

i=1

= 0, 1, . . . n, the integral transform

(VII.4.4 ) deﬁnes a bounded linear operator from L

) into L

), 1 < q <

∞, 1/r = 1/q − β, and we have

kT uk

≤ ckuk

with c = c

(q, β)M .

With this result in hand, we shall now look for a solution to (VII.4.2 )

corresponding to f , g ∈ C

∞

) of the form

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 447

v(x) =

(2π)

n/2

ix·ξ

V (ξ)dξ

p(x) =

(2π)

n/2

ix·ξ

P (ξ)dξ.

(VII.4.5 )

Replacing (VII.4.5) into (VII.4.2) furnishes the following algebraic system for

V and P :

(ξ

+ iξ

(ξ) + iξ

P (ξ) =

(ξ)

iξ

(ξ) = bg,

(VII.4.6 )

where m = 1, . . ., n. Solving (VII.4.6) for V and P delivers

(ξ) = U

(ξ) + W

(ξ)

P (ξ) = Π(ξ) + T (ξ)

(VII.4.7 )

with

(ξ) =

−ξ

(ξ

+ iξ

)

(ξ)

(ξ) = −i

bg(ξ)

Π(ξ) = i

(ξ)

T (ξ) =



+ 1



bg(ξ).

(VII.4.8 )

From (VII.4.5) and (VII.4.7), (VII.4.8) we have that a sol ution to (VII.4.2) is

given by

v(x) = u(x) + w(x)

p(x) = π(x) + τ (x)

(VII.4.9 )

with

u(x) =

(2π)

n/2

ix·ξ

U(ξ)dξ

w(x) =

(2π)

n/2

ix·ξ

W (ξ)dξ

π(x) =

(2π)

n/2

ix·ξ

Π(ξ)dξ

τ(x) =

(2π)

n/2

ix·ξ

T (ξ)dξ .

(VII.4.1 0)

Observe that u, π and w, τ are solutions of (VII.4.2) correspondi ng to f 6= 0,

g = 0 and to f = 0, g 6= 0, respectively. Since

f and bg are in S(R

), it is

448 VII Steady Oseen Fl ow in Exterior Domains

not hard to show that (VII.4.9) deﬁnes a C

∞

-solution to (VII.4 .2). Let us

now determine some L

-estimates for v and p. This will be done with the aid

of Lemma VII.4.1. In this respect, an important rol e will be played by the

function

(ξ) =

−ξ

(ξ

+ iξ

)

, (VI I.4.11)

whose properties will be investigated. Sp eciﬁcally, we have

Lemma VII.4.2 Let n ≥ 2 and let φ

be given by (VII.4.11 ) with m, k

ranging in {1, . . . , n}. Then, the assumptions of Lemma VII. 4.1 are satisﬁed:

(a) by φ

with β = 2/(n + 1);

(b) by ξ

with β = 1/(n + 1) and ` ∈ {1, . . ., n};

with β = 0;

(d) by ξ

with β = 0 and s, ` ∈ {1, . . ., n}.

Finally, if n = 2, for all `, k ∈ {1, 2} the assumptions of Lemm a VII.4.1 are

satisﬁed:

(e) by φ

with β = 1/2;

(f) by ξ

with β = 0.

Proof. Clearly, φ

and the product of φ

with any pro duct of the variables

satisfy the regularity assumptions of Lemma VI I .4.1. Moreover, for all

`, m, k = 1, . . . , n it is immediately seen that

|ξ

· ... · |ξ



∂

∂ξ

. . .∂ξ



≤ c

|ξ|

+ |ξ

for some c

= c

(n) where κ

is zero or one, κ =

i=1

= 0, 1, . . ., n, a nd

therefore to show assertion (a) it is enough to show that

(|ξ

| · ... · |ξ

2/(n+1)

|ξ|

+ |ξ

≤ c

(VII.4.1 2)

with c

= c

(n). Now, by a repeated use of Young’s inequality (II.2.7)

(|ξ

| · . . . ·|ξ

2/(n+1)

≤ a

[|ξ

1/2

+ (|ξ

| · . . . · |ξ

1/(n−1)

]

≤ a

(|ξ

| + |ξ|

with a

= a

(n), i = 1, 2, and so (VII.4.1 2) follows. Likewise, we can show

assertions (b) and (d). Furthermore, observing that

|ξ

· ... · |ξ



∂

(ξ

)

∂ξ

. . .∂ξ



≤ c

|ξ

|ξ|

+ |ξ

≤ c

with c

= c

(n), property (c) f ollows. To prove the last part of the lemma we

notice that if n = 2 from (VII.4.11),

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 449

(ξ) =

(ξ

+ iξ

)

(ξ) =

−ξ

(ξ

+ iξ

)

and therefore, for κ

= 0, 1, i = 1, 2 and k = 1, 2, we have

|ξ



∂

∂ξ



≤ c

|ξ

|ξ|(|ξ|

+ |ξ

, κ = κ

+ κ

Since

(|ξ

||ξ

1/2

|ξ

|ξ|(|ξ|

+ |ξ

≤ 1,

assertion (e) follows. Finall y, from the inequality

|ξ



∂

(ξ

)

∂ξ



≤ c

|ξ

|ξ|

+ |ξ

≤ c

(f) is proved and the proo f of the lemma . ut

Let us begin to estimate u and π. From (VII.4.8)

and (VII.4.10), with

the help of Lemma VII.4.1 and Lemma VII.4.2 (c), (d), it follows at once that



∂u

∂x



≤ ckfk

(VII.4.1 3)

and

|u|

2,q

≤ ckf k

. (VII.4.14 )

Also, observing that for all s, k = 1, . . . , n the function ξ

/ξ

satisﬁes the

assumptions of Lemma VII.4.1 with β = 0, we have

|π|

1,q

≤ ckf k

From (VII. 4.13), (VII. 4.14) and from this last inequality we conclude



∂u

∂x



+ |u|

2,q

+ |π|

1,q

≤ ckfk

, 1 < q < ∞, (VII.4.15)

with c = c(n, q). In the case of plane ﬂow (n = 2), we a re able to obtain a

sharper estimate on the com ponent u

of the veloci ty ﬁeld. Sp eciﬁcally, from

(VII.4.9 ), (VII.4.7), Lemma VII.4 .1, and Lemma VII.4.2(f), we recover

1,q

≤ ckfk

, 1 < q < ∞, (VII.4.16)

which, along with (VII.4.14), then furnishes

450 VII Steady Oseen Fl ow in Exterior Domains

1,q



∂u

∂x



+ |u|

2,q

+ |π|

1,q

≤ ckfk

, 1 < q < ∞. (VII.4.17)

Other estima tes can be obtained by suitably restricting the range of values

of q. To this end, assume 1 < q < n + 1; we then o bta in from Lemma VII.4.1

and Lemm a VII.4.2(b)

|u|

1,s

≤ ckfk

, s

(n + 1)q

n + 1 − q

, 1 < q < n + 1, (VII.4.18 )

where c = c(n, q). In the case of plane ﬂow, n = 2, in addition to (VII.4.18),

from Lemma VII.4.2(e) we derive

2q/(2−q)

≤ ckf k

, 1 < q < 2, (n = 2). (VII.4.19)

Finally, assuming 1 < q < (n + 1)/2, from Lemma VII.4.1 and Lemma

VII.4.2(a) we ﬁnd

kuk

≤ ckf k

, s

(n + 1)q

n + 1 − 2q

, 1 < q <

n + 1

. (VII.4.20)

Let us now estimate the pair w, τ . Observing that

h = iξ

h, (VII.4.2 1)

and recalling that the function ξ

/ξ

satisﬁes the assumptions of Lemma

VII.4.1 with β = 0, we at once obtain

|w|

1,r

≤ckgk

, 1 < r < ∞

|w|

2,r

≤c|g|

1.r

, 1 < r < ∞.

(VII.4.2 2)

Likewise,

|τ |

1,r

≤ ckgk

1,r

, 1 < r < ∞. (VII.4.23)

Moreover, it is simple to show that ξ

/ξ

, k = 1, . . ., n, satisﬁes the assump-

tions of Lemma VII.4.1 with β = 1/n and so

kwk

nr /(n−r)

≤ ckgk

, 1 < r < n. (VII.4.24)

Thus, from (VII.4.15), (VII.4.22), and (VII.4.23), it follows that



∂v

∂x



+ |v|

2,q

+ |p|

1,q

≤ c(kf k

+ kgk

1,q

), 1 < q < ∞, (VII.4.25)

and, from (VII.4.16), (VII.4.22), and (VII.4.23),

1,q



∂v

∂x



+ |v|

2,q

+ |p|

1,q

≤ c(kf k

+ kgk

1,q

), 1 < q < ∞, (n = 2).

(VII.4.2 6)

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 451

Remark VI I.4.1 Estimates (VII.4.25) and (VII.4.26) have no analogue in

the Stokes problem, because of the presence of the terms in

∂v

∂x

and v

. Just

these estima tes, from the point of view of the L

-approach, make the diﬀerence

between Oseen and Stokes approximations. 

If we restrict the values of q we can obtain other estimates for v. Speciﬁ-

cally, from (VII.4.18) and (VII.4.22) we obtain

|v|

1,s

≤ c(kf k

+ kgk

), s

(n + 1)q

n + 1 − q

, 1 < q < n + 1.

On the other hand, by the embedding Theorem II.3.1, it easily follows that

kgk

≤ ckgk

1,q

, s

(n + 1)q

n + 1 −q

, 1 < q < n + 1,

and so, in particular,

|v|

1,s

≤ c(kf k

+ kgk

1,q

), s

(n + 1)q

n + 1 − q

, 1 < q < n + 1. (VII.4.27)

If n = 2, from (VII.4.19) and (VII.4.24) and from the embedding TTheorem

II.3.1, we have

2q/(2−q)

≤ c(kf k

+ kgk

1,q

), 1 < q < 2, (n = 2) . (VII.4.2 8)

Finally, if 1 < q < (n +1)/2 , we choose in (VII.4.24) the exponent r such tha t

nr/(n − r) = q(n + 1)/(n + 1 − 2q) to obtain

kwk

≤ ckgk

, r

n(n + 1)q

n(n + 1 − q) + q

, 1 < q <

(n + 1)

(Notice that r

< n, since q < (n+1)/2.) Again using the embedding Theorem

II.3.1 on the right-hand side of this relation yields

kwk

≤ ckgk

1,q

which together with (VII.4.20 ), in turn, implies

kvk

≤ c(kf k

+ kgk

1,q

), s

(n + 1)q

n + 1 − 2q

, 1 < q <

(n + 1)

. (VII.4.29)

The results obtained so far can be imm ediately extended alo ng the follow-

ing two directions. First of all, estimates (VII.4.25)–(VII.4.29) can be gener-

alized to derivatives of arbitrary order, by operating with D

on both sides

of (VII.4.10) and then using (VII.4.21). Secondly, f and g can be merely as-

sumed to be in W

m,q

) and W

m+1,q

), respectively. In f act, we may

use a standard argument of the type employed in Section IV.2 fo r the Stokes

problem along with the inequalities just derived (and those for higher-order

derivatives) to establi sh existence of solutions to (VII.4.2) and related esti-

mates under the above-stated larger assumptions on f and g. Corresponding

results for system (VII.4.1) can then be obtained via transformation (VII.4.3).

The ﬁrst part of the following theorem is then acquired.

452 VII Steady Oseen Fl ow in Exterior Domains

Theorem VII.4.1 Given

f ∈ W

m,q

), g ∈ W

m+1,q

), m ≥ 0, 1 < q < ∞,

there exists a pair of functions v, p with

v ∈ W

m+2,q

), p ∈ W

m+1,q

) , f or any R > 0,

and satisfying a .e. the nonhomogeneous Oseen system (VII.4.1). Moreover,

for a ll integers ` ∈ [0, m], the quantities



∂v

∂x



`,q

, |v|

`+2,q

, |p|

`+1,q

are ﬁnite and satisfy the estima te



∂v

∂x



`,q

+ |v|

`+2,q

+ |p|

`+1,q

≤ c (R|f|

`,q

+ |g|

`+1,q

+ R|g|

`,q

) . (VII.4.30)

If n = 2,

`+1,q

is also ﬁnite and it holds that

R|v

`+1,q

+ R



∂v

∂x



`,q

+ |v|

`+2,q

+ |p|

`+1,q

≤ c (R|f|

`,q

+ |g|

`+1,q

+ R|g|

`,q

) .

(VII.4.3 1)

If 1 < q < n + 1, |v|

`+1,s

is ﬁnite, s

= (n + 1)q/(n + 1 − q), and

1/(n+1)

|v|

`+1,s

+ R



∂v

∂x



`,q

+ |v|

`+2,q

+ |p|

`+1,q

≤ c (R|f|

`,q

+ |g|

`+1,q

+ R|g|

`,q

) .

(VII.4.3 2)

If n = 2 and 1 < q < 2, |v

`,2q/(2−q)

is also ﬁnite and we have

R|v

`,2q/(2−q)

+ R|v

`+1,q

+ R

1/3

|v|

`+1,3q/(3−q)

+ R



∂v

∂x



`,q

+|v|

`+2,q

+ |p|

`+1,q

≤ c (R|f|

`,q

+ |g|

`+1,q

+ R|g|

`,q

) .

(VII.4.3 3)

Furthermore, if 1 < q < (n + 1)/2, |v|

`,s

is ﬁnite where s

= (n + 1)q/(n +

1 − 2 q), and

2/(n+1)

|v|

`,s

+ R

1/(n+1)

|v|

`+1,s

+ R



∂v

∂x



`,q

+|v|

`+2,q

+ |p|

`+1,q

≤ c (R|f|

`,q

+ |g|

`+1,q

+ R|g|

`,q

) ,

(VII.4.3 4)

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 453

so that, in particular, if n = 2, from (VII.4.33) and (VII.4.34) it follows that

R|v

`,2q/(2−q)

+ R|v

`+1,q

+ R

2/3

|v|

`,3q/(3−2q)

1/3

|v|

`+1,3q/(3−q)

+ R



∂v

∂x



`,q

+ |v|

`+2,q

+ |p|

`+1,q

≤ c (R|f |

`,q

+ |g|

`+1,q

+ R|g|

`,q

) .

(VII.4.3 5)

Here the constants c

, i = 1, 2, 3, depend only on n, `, and q. Finally, if w, τ

is another solution to (VII.4.1) corresponding to the same data f , g with



∂w

∂x



`,q

, |w|

`+2,q

ﬁnite, for some ` ∈ [0, m], then



∂

∂x

(w − v)



`,q

≡ |w − v|

`+2,q

≡ |τ −p|

`+1,q

≡ 0.

Proof. We have to show the uniqueness part only. In view of Theorem VII.1.1,

we have (w − v), (τ − p) ∈ C

∞

). Thus, letting z = D

(w − v), s =

(τ −p), with |α| = `, we derive, in particular, tha t z, s is a smooth solution

to the following system in R

∆z + R

∂z

∂x

= ∇s

∇ · z = 0.

(VII.4.3 6)

Using (VII.4.36)

in (VII.4.36)

we ﬁnd that s is harmonic in the whole space

and, since ∇s ∈ L

), from Exercise I I.11.11 we have ∇s ≡ 0, namely,

|∇s|

`,q

≡ |τ − p|

`+1,q

≡ 0. (VII.4.37)

From this and (VII.4.36)

we infer

∆z + R

∂z

∂x

= 0, (VII. 4.38)

where z is any component of z. Since D

z ∈ L

), from the following

Exercise VII. 4.1 we conclude

`,q

≡ |w − v|

`+2,q

≡ 0. (VII.4.39)

Relations (VII.4. 37)–(VII.4.39) complete the proof of the theorem. ut

The assumptions on w can be weakened. Weaker assumptions, however, would

be unessential for our aims.

454 VII Steady Oseen Fl ow in Exterior Domains

Exercise VII.4.1 Let z be a function from C

) ∩ L

), r ∈ (1, ∞), satisfy-

ing (VII.4.38). Show z ≡ 0. Hint (Rionero & Galdi 1979): Multiply both sides of

(VII.4.38) by ϕ

z|z|

r−2

, where ϕ

is a positive C

∞

-function in R

that is one in

and zero in B

, and satisﬁes |D

| ≤ cR

−|α|

, |α| = 1, 2. Integrating by parts

one arrives at

|z|

r−2

|∇z|

≤ c

(|∆ϕ

| + |∇ϕ

|) |z|

, (∗)

for some c

independent of R. The result then follows by letting R → ∞ in ( ∗).

The last part of this section is devoted to show existence and uniqueness

of a q-weak solution to (VII.4.2) (or, what amounts to the same thing, to

(VII.4.1 )). By a q-weak solution t o (VII.4.2) we mean a ﬁeld v satisfying the

conditions

(i) v ∈ D

1,q

);

(ii) (∇v, ∇φ) − (

∂v

∂x

, φ) = −[f , φ], for all φ ∈ D(Ω);

(iii) (v, ∇ϕ) = −(g, ϕ), for all ϕ ∈ C

∞

(VII.4.4 0)

In view of Lemma VII.1.1, we can associate to v a pressure ﬁeld p ∈ L

loc

(R)

such that

(∇v, ∇ψ) − (

∂v

∂x

, ψ) −(p, ∇ ·ψ) = −[f, ψ], for all ψ ∈ C

∞

As before, we begin to take f , g ∈ C

∞

). Consequently, (VII.4.8), (VII.4.9),

(VII.4.1 0) is a C

∞

-solution to (VII.4.2). We now observe that f and g can be

written in a divergence form; that is,

(x) = D

(x)

g(x) = D

(x),

(VII.4.4 1)

where F = F (x) and G = G(x) are second-order tensor and vector ﬁelds,

respectively, satisfying

|f|

−1,q

≤ kF k

≤ c

|f |

−1,q

|g|

−1,q

≤ kGk

≤ c

|g|

−1,q

|G|

1,q

≤ c

kgk

for a ll q ∈ (1, ∞) (VII.4.42)

Actually, we may choose

= D

(E ∗ f

)

= D

(E ∗ g)

with E the Laplace fundamental solution (II.9.1), a nd then use the result of

Exercise II.11. 9. Notice that

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 455

= iξ

(VII.4.4 3)

We begin to g ive an estimate for the pair u, π. From (VII. 4.10), (VII.4.8)

1,3

and (VII.4.43), with the aid of Lemma VII.4.1 and Lemma VII.4.2(d), it

follows that

|u|

1,q

+ kπk

≤ c

kF k

, 1 < q < ∞,

and (VII.4.42)

implies

|u|

1,q

+ kπk

≤ c

|f|

−1,q

, 1 < q < ∞. (VII.4.44)

In the case of plane ﬂow (n = 2), we can use Lemma VII.4.2(f) to obtain a

further estimate for the component u

, namely,

≤ c

kF k

, 1 < q < ∞.

Therefore, in such a case, this last relation, along with (VII.4.42)

and

(VII.4.4 4), furnishes

1,q

+ |u

1,q

+ kπk

≤ c

|f |

−1,q

, 1 < q < ∞. (VII.4.45)

If we restrict the range of values of q, we may obtain another esti-

mate. Speciﬁcally, from (VII.4. 10), (VI I.4.8)

, Lemma VII.4.1 , and Lemma

VII.4.2(b) we ﬁnd

kuk

≤ c

kF k

, s

(n + 1)q

n + 1 − q

, 1 < q < n + 1,

so that (VII.4.42)

and (VII.4.44) imply

kuk

+|u|

1,q

+kπk

≤ c

|f |

−1,q

, s

(n + 1)q

n + 1 −q

, 1 < q < n+1. (VII.4.46)

Moreover, if n = 2, inequalities (VII.4.45) a nd (VII.4.46) deliver

1,q

+ kuk

3q/(3−q)

+ |u

1,q

+ kπk

≤ c

|f |

−1,q

, 1 < q < 3. (VII.4.47)

We shall next estimate the pair w, τ deﬁned by (VII.4.10)

2,4

and (VII.4.8)

2,4

respectively, with g given by (VII.4.41)

. Actually, recalling that ξ

/ξ

k, s = 1, . . ., n, satisﬁes the assumptions of Lemma VII.4.1 with β = 0 we

at once obtain

kwk

≤c

|g|

−1,r

|w|

1,r

≤c

kgk

kτk

≤c

(kgk

+ |g|

−1,r

) .

1 < r < ∞ (VII.4.48)

On the other hand, from the embedding Theorem II.3.1 we ﬁnd