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

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836 XII Two-Dimensional Flow in Exterior Domains

|x|

(1+ε)/2

F (u) ∈ L

(Ω

) , R > R

, (XII.4.39)

so that, by (XII.4.37)–(XI I.4.39), we conclude

|x|

(1+ε)/2

− 1) ∈ L

(Ω

) .

Let us denote by L

(Ω

) the (weighted) space of functions g with the property

|x|

(1+ε)/2

g ∈ L

(Ω

). Observing that u

−1 = v

−1−v

+2i v

, we deduce:

(i) −v

+ (v

+ 1)v

= g, and (ii) v

−1 − v

= h, with g, h ∈ L

(Ω

). Thus,

recalling that: (iii) v

(x) + 1 = o(1) as |x| → ∞, from (i) it follows, for

suﬃciently large |x|,

(x)| ≤ |g(x)| + |v

(x) + 1||v

(x)| ≤ |g(x)|+

(x)|,

that is, v

∈ L

(Ω

Replacing this information back in (ii), we ﬁnd v

−

1 = f, with f ∈ L

(Ω

). Since 2(v

+ 1) = (v

+ 1)(v

+ 1) − v

+ 1, by (iii)

we recover, for suﬃciently large |x|,

2|v

(x) + 1| ≤ | f(x)| + |v

(x) + 1||v

(x) + 1| ≤ |f(x) | + |v

(x) + 1|

that is v

+ 1 ∈ L

(Ω

), a nd the proof of the theorem is completed.

XII.5 Existence and Uniqueness of Solutions for Small

Data and v

∞

6= 0

Objective of this section is to show that the two-dimensional exterior probl em

admi ts a unique solution –at least when the velocity v

∞

is not zero and the

data are suﬃciently small.

The leading idea is the same we used in similar circumstances, namely,

to couple suitably the L

-estimates derived for the linearized approximation

together with a contraction mapping argument. It is important to empha -

size, however, that this procedure i s not going to work unless v

∞

6= 0 and,

even in this situation, its success is by no means evident a priori, as we are

about to illustrate. To begin, let us take ﬁrst v

∞

= 0. As we know from the

linearized Stokes theory of Section V.3 and Section V.5, in such a case only

two types of existence results with corresponding L

-estimates are available:

those of Theorem V.4 .6 and those of Theorem V.5.1. Now, by those of Theo-

rem V.4.6 we are not able to control the behavior of the solution at inﬁnity,

while the results of Theorem V.5.1 require for their validity a (necessary a nd

suﬃcient) compatibility condition on the data, and we cannot use them to

prove existence unless we show that such a condition is also necessary in the

nonli near case. We may then conclude that the L

theory developed for the

Recall that v is C

∞

(Ω

XII.5 Existence and Uniqueness of Solutions for Small Data and v

∞

6= 0 837

Stokes problem, as it stands, cannot be used to obtain existence for the ful l

nonli near Navier–Stokes problem. In fact, existence o f solutions of the exterior

two-dimensional Navier–Stokes problem corresponding to v

∞

= 0 remains an

open question, even for smal l data.

On the other hand, the L

theory for the linearized Oseen problem de-

rived in Theorem VII.5.1 does not suﬀer from these drawbacks and can be

used, at least in principle, to show existence to the nonlinear problem when

∞

6= 0, v

∞

= e

, say. However, employing these estimates, together with a

contraction-mapping technique, we have to face other problems. Actually, we

wish to ﬁnd a ﬁxed poi nt (in a subset X of an appropriate Banach space) of

the mapping

L : u ∈ X → L(u) = v ∈ X

where v solves the problem

∆v + R

∂v

∂x

= Ru · ∇u + ∇π + f

∇ · v = 0











in Ω

v = v

∗

+ e

at ∂Ω

lim

|x|→∞

v(x) = 0.

(XII.5.1 )

In the limit of small data, i.e., R → 0, as is usually requested by a contraction

argument, there will be a competition between the linear term

∂v

∂x

(XII.5.2 )

and the nonlinear one

Ru · ∇u. (XII.5.3)

If, in the range of vanishing R, the contribution of the former were negligible

with respect to that of the latter, i t would be very unlikely to prove existence,

because the linear part in (XII.5.1) would then approach the Stokes system

for which, as we noticed, the procedure is not working. Fortunately, what

happ ens is that (XII.5.2) prevails on (XII.5.3) and the ma chinery produces

nonli near existence. Nevertheless, the proof o f this fact is by no means trivial

and, in fact, i t relies on the following two crucial circumstances:

(a) The validity of suitable, new a priori estimates for solutions to the

linearized Oseen problem where, unlike those derived in Theorem VII.7.1

(cf. (VII.7 .29)), the constant c entering the estimates themselves does not

depend on R ∈ (0, 1]; cf. Lemma XII.5.1. Because of the Stokes paradox, the

new estimates have to be weaker than (VII.7.29); cf. Remark VII.7.2.

See t he Notes for this Chapter regarding some existence results when v

∞

= 0.

838 XII Two-Dimensional Flow in Exterior Domains

(b) The component v

of the velocity ﬁeld v in the solution to the Oseen

problem, presents no “wake region” a nd, as a consequence, it has at large

distances a behavior “better” than that exhibited by the component v

For simplicity, we shall restrict ourselves to the case when the domain

Ω is exterior to a single “body” (the closure of a bounded, simply-connected

domain), leaving to the reader the task of considering more general situa tions.

In addition, in order to render the notation less heavy, throughout this section

the Reynolds number R will be denoted by λ.

We next introduce som e other notations that will be frequently used in the

sequel. For q ∈ (1, 6/5], we indicate by C

the class of those vector functions

u = (u

, u

) deﬁned in Ω such that:

∈ L

2q/(2−q)

(Ω) ∩ D

1,q

(Ω)

u ∈ L

3q/(3−2q)

(Ω) ∩ D

2,q

(Ω).

For u ∈ C

and λ > 0, we put

hui

λ,q

≡ λ(ku

2q/(2−q)

+ |u

1,q

) + λ

2/3

kuk

3q/(3−2q)

+ λ

1/3

|u|

1,3q/(3−q)

(XII.5.4 )

and, as usual, we write hui

λ,q,Ω

when we need to specify the domain where

the norm (XII.5.4) is deﬁned.

Remark XI I.5.1 For u ∈ C

, it holds that

lim

|x|→∞

u(x) = 0 uniformly.

Actually, by Theorem II.6.1, u ∈ D

1,2q/(2−q)

(Ω). Since 2q/(2 − q) > 2 and

u ∈ L

3q/(3−2q)

(Ω), the property follows from Theorem II.9.1. If u has only

ﬁnite norm (XII. 5.4), then

lim

|x|→∞

2π

|u(|x|, θ)|dθ = 0. (XII.5.5)

In fact, if q ∈ (1, 6/5) then u ∈ D

1,r

(Ω) for some r < 2 and so from Lemma

II.6.3 and the condition u ∈ L

3q/(3−2q)

(Ω) we ﬁnd (XII.5.5). If q = 6/5, we

have u ∈ D

1,2

(Ω) ∩ L

(Ω), and we proceed as follows. Set

f(r) =

2π

|u(r, θ)|

dθ, r = |x|.

Then there exi sts a sequence {r

} ⊂ R

accumulating at inﬁnity such that

lim

→∞

f(r

) = 0.

For all suﬃciently large r we may ﬁnd n ∈ N such that

XII.5 Existence and Uniqueness of Solutions for Small Data and v

∞

6= 0 839

f(r) = f(r

) + 2

2π

u(ρ, θ) ·

∂u(ρ, θ)

∂ρ

dρdθ.

Applying the H¨older inequality on the right-hand side of this relation, it fol-

lows that

|f(r)| ≤ |f(r

)|+ 2kuk

6,Ω

|u|

1,2,Ω



∞

−2

dρ



1/3

which again proves (XII.5.5). 

We shall now give some preparatory results. Our ﬁrst objective is to prove

a fundamental estimate fo r the linearized Oseen problem corresponding to

zero body force.

Lemma XII.5.1 Let Ω ⊂ R

be an exterior domain of class C

and let

∗

∈ W

2−1/q,q

(∂Ω), 1 < q ≤ 6/5.

Then, for any λ > 0 there is a unique solution to the Oseen problem

∆u + λ

∂u

∂x

= ∇π

∇· u = 0











in Ω

u = u

∗

at ∂Ω

lim

|x|→∞

u(x) = 0

(XII.5.6 )

such that u ∈ C

, π ∈ D

1,q

(Ω). Moreover, if q ∈ (1, 6/5) there is a λ

> 0

such that for all λ ∈ (0, λ

]

hui

λ,q

≤ cλ

2(1 −1/q )

|log λ|

−1

∗

2−1/q,q(∂Ω)

(XII.5.7 )

with c = c(Ω, q, λ

Proof. From Theorem VII.5.1 we know that there is a unique solution to

problem (XII.5.6) with hui

λ,q

ﬁnite and such that

u ∈ D

2,q

(Ω

), π ∈ D

1,q

(Ω

). (XII. 5.8)

On the other hand, from Theorem IV.4.1 and Theorem IV.5.1 we obtain

kuk

2,q,Ω

+ kπk

1,q,Ω

≤ c



λkD

q,Ω

+ kuk

1,q,Ω

+kπk

q,Ω

+ ku

∗

2−1/q,q(∂Ω)



(XII.5.9 )

Since, by Theorem VII.5.1 and (VII.7.14), u and π satisfy for all R > δ(Ω

)

kuk

2,Ω

+ kπk

2,Ω

+ |u|

1,2

≤ c

(1 + λ)ku

∗

1/2,2(∂Ω)

, (XII.5.1 0)

840 XII Two-Dimensional Flow in Exterior Domains

from (XII.5.9), (XII.5.10) it follows that

kuk

2,q,Ω

+ kπk

1,q ,Ω

≤ c

(ku

∗

2−1/q,q(∂Ω)

+ ku

∗

1/2,2(∂Ω)

). (XII.5.11)

However, the trace Theorem II.4.4 implies that

∗

1/2,2(∂Ω)

≤ cku

∗

2−1/q,q(∂Ω)

and from (XII.5 .11) we conclude that

kuk

2,q,Ω

+ kπk

1,q,Ω

≤ c

∗

2−1/q,q(∂Ω)

(XII.5.1 2)

with c

independent of λ ∈ (0, B], arbi trary B > 0. Thus, in particular,

(XII.5.9 ) and (XII.5.13) tell us

u ∈ C

, π ∈ D

1,q

(Ω),

completing the proof of the ﬁrst part of the lemma. We now pass to the proof

of (XII.5.7). W ithout l oss o f generality, we assume that Ω

⊂ B

1/2

, with the

origin of coordinates taken in

◦

Ω

. From the embedding Theorem II. 3.4,

2q/(2−q),Ω

+ |u

1,q,Ω

+ kuk

3q/(3−2q),Ω

+ |u|

1,3q/(3−q),Ω

≤ c

|u|

2,q,Ω

(XII.5.1 3)

and so, since 2(1 − 1/q) < 1/3 f or q ∈ (1, 6/5), from (XII.5.12), (XII.5.13),

and all λ ∈ (0, 1] we ﬁnd for some c

independent of λ

hui

λ,q,Ω

≤ c

2(1 −1/q ) +ε

∗

2−1/q,q(∂Ω)

(XII.5.1 4)

where ε = 1/3 − 2(1 − 1/q) > 0. Let us denote by E(x; λ) the Oseen tensor

corresponding to the Reynolds number λ. From Exercise VII.3.5 we know that

E ob eys the homogeneity condition

E(x; λ) = E(λx; 1). (XII.5.15)

Moreover, from the representation Theorem VII.6.2 we deduce for j = 1, 2

that

= −T (u) · E

(x; λ)

∂Ω

∗

(z) ·T [E

(x −z; λ), e

(x − z)] · n(z)dσ

−λ

∂Ω

∗

(z) · E

(x − z; λ)n

(z)dσ

−

∂Ω

(x − z; λ) − E

(x; λ)] · T (u, π) · n(z)dσ

(XII.5.1 6)

with

XII.5 Existence and Uniqueness of Solutions for Small Data and v

∞

6= 0 841

T (u) =

∂Ω

T (u, π) · n

and

= (E

, E

)

= (E

, E

(XII.5.1 7)

Recalling that Ω

⊂ B

1/2

from (XII.5.16) and the mean value theorem applied

to D

(x − z; λ) −E

(x; λ)), α = 0, 1, we derive for all x ∈ Ω

(x)| ≤ |T (u)||E

(x; λ)|

+D{λ sup

z∈Ω

1/2

(x − z; λ)|

+ sup

z∈Ω

1/2

[|e(x − z)|+ |∇

(x − z; λ)|]}

|∇u

(x)| ≤ |T (u)||∇E

(x; λ)|+ D{λ sup

z∈Ω

1/2

|∇

(x − z; λ)|

+ sup

z∈Ω

1/2

[|∇

e(x − z)| + |D

(x − z; λ)|]}

(XII.5.1 8)

where

D = k∇uk

1,∂Ω

+ kπk

1,∂Ω

+ ku

∗

1,∂Ω

. (XII.5.19)

Taking into account (XII.5.15), from (XII.5.18)

we ﬁnd, with y = λx,

(x)| ≤ |T (u)||E

(y; 1)| + λD{ sup

z∈Ω

1/2

[|E

(y − λz; 1)|

+ |e(y − λz)| + |∇

(y − λz; 1)|]}

≤ |T (u)||E

(y; 1)| + λD{ sup

|z|≤λ/2

[|E

(y − z; 1)|

+ |e(y − z)| + |∇

(y − z; 1)|]}

and so

t,Ω

≤ 2

−2

{|T (u)|

(y; 1)k

t,R

+ λ

|y|≥λ

{sup

|z|≤λ/2

[|E

(y − z; 1)|

+ |e(y − z)| + |∇

(y − z; 1)|]}

dy}.

(XII.5.2 0)

To estima te the integral on the right-hand side of (XII.5.20) we observe that

|y|≥λ

sup

|z|≤λ/2

(y − z; 1)|

dy ≤

2≥|y|≥λ

sup

|z|≤λ/2

(y − z; 1)|

|y|≥2

sup

|z|≤1/2

(y −z; 1)|

dy.

(XII.5.2 1)

842 XII Two-Dimensional Flow in Exterior Domains

From (VII. 3.36) we have

|E(y − z; 1)| ≤ c(|log |y −z|| + 1), |y|, |z| ≤ 2

and since

|z| ≤ λ/2, |y| ≥ λ implies |y − z| ≥

|y|, (XII.5.22)

it follows that

2≥|y|≥λ

sup

|z|≤λ/2

(y − z; 1)|

dy ≤ c

(XII.5.2 3)

with c

independent of λ ∈ (0, 1]. Furthermore, since by the mean value

theorem,

|(E

(y − z) − E

(y))| = | z

∂E

(y − βz)

∂z

|, β ∈ (0, 1),

from (VII.3.37) and (VII.3.45) it follows for all | y| suﬃciently large (≥ |y

say) that

|y|≥|y

sup

|z|≤1/2

(y − z; 1)|

dy ≤ c

|y|≥|y

(|E

(y; 1)|

+ |y|

−tr

)dy

where

(

1 if j = 1

3/2 if j = 2.

From the local regularity of E, and from (VII.3.42), (VII.3.43) we conclude

that

|y|≥2

sup

|z|≤1/2

(y −z; 1)|

dy ≤ c

(XII.5.2 4)

with c

independent of λ ∈ (0, 1] for all values of t such that

(y; 1) ∈ L

(|y| ≥ |y

|). (XII.5.25)

Collecting (XII.5.21), (XII.5. 23), and (XII.5.24), it follows that

|y|≥λ

sup

|z|≤λ/2

(y −z; 1)|

dy ≤ c

(XII.5.2 6)

with c

independent of λ ∈ (0, 1] and for all values of t for which (XII.5.25)

holds. In addition, from (VII.3.17) and (XII.5.22) we infer for all t > 2 that

|y|≥λ

sup

|z|≤λ/2

|e(y − z; 1)|

dy ≤ c



2≥|y|≥λ

|y|

−t

|y|≥2

sup

|z|≤1/2

|y − z|

−t



≤ c

(λ

2−t

+ 1).

(XII.5.2 7)

XII.5 Existence and Uniqueness of Solutions for Small Data and v

∞

6= 0 843

Likewise,

|y|≥λ

sup

|z|≤λ/2

|∇

(y − z; 1)|

dy ≤

2≥|y|≥λ

sup

|z|≤λ/2

|∇

(y −z; 1)|

dy +

|y|≥2

sup

|z|≤1/2

|∇

(y − z; 1)|

dy.

(XII.5.2 8)

From (VII. 3.36) we have

|∇E(y −z; 1)| ≤ c

|y −z|

−1

, |y|, |z| ≤ 2

and so, by (XII.5.22), it follows that

2≥|y|≥λ

sup

|z|≤λ/2

|∇

(y − z; 1)|

dy ≤ c

2≥|y|≥λ

|y|

−t

dy ≤ c

(1 + λ

2−t

(XII.5.2 9)

Also, employing the asymptotic properties of ∇E

(y; 1) (cf. (VI I.3.46) and

(VII.3.4 7)) together with the following ones on the second derivatives,

(y)| ≤ c|y|

−s

where

(

3/2 for j = 1

2 for j = 2

we are able to establish, exactly as we did for (XII.5.24), the fo llowing esti-

mate:

|y|≥2

sup

|z|≤1/2

|∇

(x −z; 1)|

dy ≤ c

(XII.5.3 0)

for those values of t such that

∇E

(y; 1) ∈ L

(|y| ≥ |y

|). (XII.5.31)

Thus, from (XII.5.28)–(XII.5.30) we recover

|y|≥λ

sup

|z|≤λ/2

|∇

(y − z; 1)|

dy ≤ c

(1 + λ

2−t

), (XII. 5.32)

with c

independent of λ a nd t satisfying (XII.5.31). From (XII.5.20),

(XII.5.2 6), (XII.5.2 7), and (XII.5.32) we ﬁnd

t,Ω

≤ c



−2/t

|T (u))| + (1 + λ

1−2/t



These bounds on D

are obtained directly by a formal diﬀerentiation of

(VII.3.37).

844 XII Two-Dimensional Flow in Exterior Domains

for all λ ∈ (0, 1], for all t for which (XII.5.25) and (XII.5.31) hold, and with

a constant c

independent of λ. Since t > 2, from thi s latter inequality, we

deduce that

t,Ω

≤ 2c



−2/t

|T (u))| + D



. (XII.5.33)

Estimate (XII.5.33) furnishes, in particular,

λku

2q/(2−q),Ω

+ λ

2/3

kuk

3q/(3−2q),Ω

≤ c



2(1 −1/q )

|T (u))| + λ

2/3



(XII.5.3 4)

with c independent of λ ∈ (0, 1].

In a completely anal ogous way, starting

with (XII.5.18)

we show

1,τ,Ω

≤ 2

τ−2

{|T (u)|

k∇E

(y; 1)k

τ,R

+ λ

|y|≥λ

{ sup

|z|≤λ/2

[|∇

(y − z; 1)|

+ |∇

e(y − z)| + |D

(y − z; 1)|]}

dy} .

Therefore, noting that, by (VI I .3.17),

|∇

e(y − z)| ≤ c

|y −z|

−2

and that by (VI I.3.21),

2≥|y|≥λ

sup

|z|≤λ/2

(y −z; 1)|

dy ≤ c

2≥|y|≥λ

|y|

−2τ

≤ c

(1 + λ

2(1−τ)

using the same procedure employed to obta in (XII.5.33), we arrive at

1,τ,Ω

≤ c



1−2/τ

|T (u)| + D



(XII.5.3 5)

for a ll values of λ ∈ (0, 1] and all values of τ such that

∇E

(y; 1) ∈ L

(|y| ≥ |y

|),

∈ L

(|y| ≥ |y

|).

Thus, from (VII.3.46), (VII.3.49) and (XII.5.35), and observing that τ > 1,

we derive, in particular,

λ|u

1,q,Ω

+ λ

1/3

|u|

1,3q/(3−q),Ω

≤ c



2(1 −1/q)

|T (u)| + λ

1/3



(XII.5.3 6)

Observe that λ ≤ λ

2/3

for 0 ≤ λ ≤ 1.

XII.5 Existence and Uniqueness of Solutions for Small Data and v

∞

6= 0 845

for some c = c(Ω, q) and with q ∈ (1, 6/5). The next step is to increase D

(deﬁned in (XII.5.19)) in terms of the boundary no rm ku

∗

2−1/q,q(∂Ω)

. Using

the trace Theorem II.4.4 we have

D ≤ c



kuk

2,q,Ω

+ kπk

1,q,Ω

+ ku

∗

2−1/q,q(∂Ω)



and so this inequality, together with (XI I.5.12), implies

D ≤ c

∗

2−1/q,q(∂Ω)

Observing that 1/3 > 2(1 − 1/q) for q ∈ (1, 6/5), from (XII.5.14), (XII.5.34 ),

and (XII.5.36) we obtain

hui

λ,q

≤ c

2(1 −1/q )



|T (u)| + λ

∗

2−1/q,q(∂Ω)



(XII.5.3 7)

with ε = 1/3 − 2(1 − 1/q) > 0. From Theorem VII.8.1 we know that there is

a λ

> 0 such that

|T (u)| ≤ c

|log λ|

−1

∗

2−1/q,q(∂Ω)

for all λ ∈ (0, λ

] and with c

= c

(Ω, q, λ

). Thus, replacing this estimate

in (XII.5.3 7), we recover (XII.5.7) with λ

= min{1, λ

}, and the lemma is

proved. ut

Remark XI I.5.2 By using the same li ne of proof, one can show tha t if q =

6/5, the solutions o f the previous lemma satisfy the weaker estimate

hui

λ,q

≤ cλ

2(1 −1/q )

∗

2−1/q,q(∂Ω)

for a ll λ ∈ (0, 1]. 

The next result shows an estimate for solutions to the Oseen problem

with zero boundary data. It is somehow weaker than that shown in Theorem

VII.7.1, cf. (VII.7.29), but with the advantage that, in the present case, the

constant c that enters the estimate can be rendered independent of λ ∈ (0, 1 ].

Lemma XII.5.2 Let Ω be as in Lemma XI I.5.1. Then, given

f ∈ L

(Ω), 1 < q < 6/5,

there is at least one solution to the Oseen problem:

∆w + λ

∂w

∂x

= ∇τ + f

∇· w = 0











in Ω

w = 0 at ∂Ω

lim

|x|→∞

w(x) = 0

(XII.5.3 8)