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

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456 VII Steady Oseen Fl ow in Exterior Domains

kGk

≤ ckGk

1,q

, s

(n + 1)q

n + 1 − q

, 1 < q < n + 1, (VII.4 .49)

and so, from (VII.4.41)

2,3

, (VII. 4.48), and (VII.4.49) it foll ows that

kwk

1,q

+ kτ k

≤ c

(kgk

+ |g|

−1,q

) , 1 < q < ∞,

kwk

≤ c

(kgk

+ |g|

−1,q

) , s

(n + 1)q

n + 1 − q

, 1 < q < n + 1.

(VII.4.5 0)

In view of (VII.4.9), (VII.4.44)–(VII.4.47), (VII. 4.50), and using the replace-

ments (VII.4.3 ) we may conclude that, for given f and g from C

∞

) prop-

erties stated previously, there exists a q-weak solution to (VII.4.1) satisfy-

ing to gether w ith the corresponding pressure p, the following estimates: if

1 < q < ∞

|v|

1,q

+ kpk

≤ c (R|f|

−1,q

+ R|g|

−1,q

+ kgk

)

Rkv

+ |v|

1,q

+ kpk

≤ c (R|f|

−1,q

+ R|g|

−1,q

+ kgk

) ( if n = 2)

(VII.4.5 1)

and if 1 < q < n + 1

1/(n+1)

kvk

+ |v|

1,q

+ kpk

≤ c (R|f |

−1,q

+ R|g|

−1,q

+ kgk

) , s

(n + 1)q

n + 1 − q

Rkv

+ R

1/3

kvk

3q/(3−q)

+ |v|

1,q

+ kpk

≤ c (R|f |

−1,q

+ R|g|

−1,q

+ kgk

) ( if n = 2)

(VII.4.5 2)

with c = c(n, q).

Starting from (VII.4.51) and (VII.4.52), we may use a now standard den-

sity argument to extend the above results to the case when f and g merely

satisfy the assumptions

f ∈ D

−1,q

), g ∈ L

) ∩D

−1,q

However, to do this we need a result that ensures us that we can approximate

g by functions from C

∞

) in the norm of L

) ∩D

−1,q

). This is the

content of the following.

Lemma VII.4.3 Let

g ∈ L

) ∩ D

−1,q

), 1 < q < ∞.

Then, for any η > 0 there exists g

(η)

∈ C

∞

) such that

(η)

−g|

−1,q

+ kg

(η)

− gk

< η.

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole Space 457

Proof. Consider the problem

∆Ψ = g in R

. (VII. 4.53)

In view of the assumption made on g, by Exercise II.11.9 there exist two (a

priori distinct) solutions Ψ

and Ψ

to (VII.4.53) such that

|Ψ

2,q

≤ ckgk

|Ψ

1,q

≤ c|g|

−1,q

(VII.4.5 4)

It is not diﬃcult to show that

(Ψ

− Ψ

) ≡ 0. (VII.4.55)

In fact, the diﬀerence Ψ = Ψ

−Ψ

satisﬁes

∆Ψ = 0 in R

. (VII. 4.56)

We may now represent Ψ by means of (V.3.14). Taking into account (VII.4.56)

and that H

(R)

is of bounded support, this latter leads, in particular, to the

following

Ψ(x) = −

(R)

(x − y)D

Ψ(y)dy . (VII.4.57)

From (VII. 4.57) we infer

Ψ(x) = −

(R)

(x − y)D

(y)dy

(R)

(x − y)D

(y)dy,

and so, by the H¨older inequality, we obta in

Ψ(x)| ≤ kH

(R)

|Ψ

2,q

+ |H

(R)

1,q

|Ψ

1,q

Letting R → ∞ into this relation and using (VII.4.54) and (V.3.13) proves

(VII.4.5 5). We may then state that problem (VII.4.53) with g verifying the

assumptions of the lemma admits a unique solution Ψ ∈ D

1,q

) ∩D

2,q

)

and that this solution satisﬁes

|Ψ|

2,q

≤ ckgk

|Ψ|

1,q

≤ c|g|

−1,q

(VII.4.5 8)

Given ρ > 0, let us denote by ζ

= ζ

(x) a nonincreasing, smooth function

that equals 1 for |x| ≤ ρ and 0 for |x| ≥ 2ρ and

|∇ζ

| ≤ c/ρ. (VII.4.5 9)

458 VII Steady Oseen Fl ow in Exterior Domains

Set

u = ∇Ψ, u

ρ,ε

= ζ

∇Ψ

, g

ρ,ε

= ∇ · u

ρ,ε

, (VII.4.60 )

where Ψ

is the regularizer of Ψ. Evi dently

ρ,ε

∈ C

∞

Let us next show that g

ρ,ε

approaches g simultaneously in L

) and

−1,q

) as ρ → ∞ and ε → 0. By (VII.4.53), (VII.4.59), and (VII.4.60), by

the Minkowski inequality, property (II.2.9)

of the regularizer and by Exercise

II.3.2 we ﬁnd

ρ,ε

−gk

= k∇· u

ρ,ε

−∇ · uk

≤k∇ζ

· ∇Ψ

+ kζ

−gk

≤

|g|

−1,q

+ k(ζ

− 1)gk

+ kg

− gk

This inequality, along with (I I.2.9)

, establishes that for all η > 0 we m ay ﬁnd

= ρ

(η, g) > 0 and ε

= ε

(η, g) > 0 such that

ρ,ε

−gk

< η/2, for all ρ > ρ

, ε > ε

. (VII.4.6 1)

Furthermore, from (VII.4.53), (VI I .4.60) and the Minkowski inequali ty, we

have

ρ,ε

−g|

−1,q

≤ ku

ρ,ε

− uk

≤ |Ψ

− Ψ|

1,q

+ k(ζ

− 1)∇Ψk

and so, again by Exercise II.3.2 and (VII.4.58)

, it follows that for al l η > 0

we may ﬁnd ρ

= ρ

(η, g) > 0 and ε

= ε

(η, g) > 0 such that

ρ,ε

−g|

−1,q

< η/2, for all ρ > ρ

, ε > ε

. (VII.4.62)

The lemma then fol lows from (VII.4.61 ) and (VII.4.62). ut

We shall ﬁnally show tha t the q-generalized solutions just obtained are in

fact unique among their class of existence.

Actually, supp ose w, τ is another

solution corresponding to the same data, with

w ∈ D

1,q

), τ ∈ L

loc

The diﬀerences

z = w −v ∈ D

1,q

), s = τ −p ∈ L

loc

)

satisfy the identity

We observe that uniqueness is an immediate consequence of Theorem VII.6.2,

which wi ll be proved in Section VII.6. Here, we prefer to give another proof,

which makes all treatment self-contained.

This assumption on τ is a consequence of t hat made on w, see Lemma VII.1.1.

Notice that we need no global summability assumption on τ.

VII.5 Existence of Plane Flow in Exterior Domains 459

(∇z, ∇ψ) − R(

∂z

∂x

, ψ) = (s, ∇ ·ψ) (VII.4.63)

for all ψ ∈ C

∞

). From Theorem VI I.1.1 we deduce z, s ∈ C

∞

) and

so, in particular, using (VII.4.63), we ﬁnd

∆z + R

∂z

∂x

− ∇s = 0

∇ · z = 0











in R

z ∈ D

1,q

(VII.4.6 4)

Using Theorem V.5.3 (with f ≡ −R(∂z/∂x

)) it follows that

z ∈ D

2,q

), s ∈ D

1,q

Operating with ∇· o n both sides of (VII.4.64)

we ﬁnd that s is harmonic

throughout R

and, therefore, by Exercise II.11.11 we deduce ∇ s ≡ 0 in

. Relation (VII. 4.64) then furnishes that each component z of z satisﬁes

(VII.4.3 8) and si nce z ∈ D

1,q

) we may use Exercise VII.4.1 to deduce

z = 0 in D

1,q

) and uniqueness is proved.

The results just shown are summarized in the foll owing.

Theorem VII.4.2 Given

f ∈ D

−1,q

), g ∈ L

) ∩ D

−1,q

), 1 < q < ∞,

there exists at least one q-generalized solution v to (VII.4.1). Moreover, the

pressure ﬁeld p associ ated to v satisﬁes

p ∈ L

)

and v, p verify the estimate (VII.4.51). Also, if 1 < q < n + 1, then

v ∈ L

), s

(n + 1)q

n + 1 − q

with

∈ L

), if n = 2,

and estimate (VII. 4.52) holds. Finally, if w is another q-generali zed solution to

(VII.4.1 ) corresponding to the same data f and g, then w ≡ v +c

, τ ≡ p+c

for some constants c

, i = 1, 2, where τ is the pressure ﬁeld associated to w

by Lemma VII.1.1. If q < n + 1, we may take c

= 0.

VII.5 Existence of Generalized Soluti ons for Plane Flows

in Exterior Domains

In the previous section we proved, among other things, existence of pla ne ﬂow

in the whole space tending to a prescrib ed value at inﬁnity. In this section

460 VII Steady Oseen Fl ow in Exterior Domains

we shall establish the same result in the more general case when the relevant

region of ﬂow is an exterior domain. This is in sharp contrast with the Stokes

approximation, where we know that the problem is resolvable if and only if

certain restrictions are imposed on the data, see Section V. 7. However, as we

already observed in Remark VII.2.1, to show existence we cannot use sic et

simpliciter the technique of Theorem VII.1.1, for then we would not be able

to control the behavior of the solution at large distances. Consequently, we

have to employ a diﬀerent approach.

To this end, following Finn & Smith (19 67a), we begin to consider the

modiﬁed problem

∆v + R

∂v

∂x

−εv = ∇p + Rf

∇ · v = 0











in Ω

v = v

∗

at ∂Ω

lim

|x|→∞

v(x) = 0

(VII.5.1 )

with ε ∈ (0, 1] and prove for it existence of solutions and suitable L

-estimates

for any value of ε. Successively, using such estimates, we show that in the limit

ε → 0 these solutions converge in a well-deﬁned sense to a generalized solution

of the original problem (VII.0.2), (VII.0.3).

The above assertions will be proved through several steps. First of all we

consider existence of a solution to the following nonhomogeneous approximat-

ing system in R

∆u + R

∂u

∂x

−εu = ∇π + RF

∇ · u = g

(VII.5.2 )

Speciﬁcally, we have

Lemma VII.5.1 Let F ∈ L

), g ∈ W

1,q

), 1 < q < 3/2. Then for all

ε ∈ (0, 1] there exists a solution u

, π

to (VII .5.2) such that

∈ W

2,q

) ∩ D

1,3q/(3−q)

) ∩ L

3q/(3−2q)

)

∈ D

1,q

)

∈ L

2q/(2−q)

) ∩ D

1,q

)

∂u

∂x

∈ L

)

and satisfying the estimate

VII.5 Existence of Plane Flow in Exterior Domains 461

Rku

2q/(2−q)

+R|u

1,q

+ R

2/3

3q/(3−2q)

+ R

1/3

1,3q/(3−q)



∂u

∂x



+ |u

1,q

+ |π

1,q

≤ c (R|f|

+ |g|

1,q

+ Rkgk

) ,

(VII.5.3 )

where c = c(q). Moreover, if w, τ are such that

(a) w ∈ W

1,2

), τ ∈ L

loc

);

(b) (∇w, ∇ψ) −R(

∂w

∂x

, ψ) + ε(w, ψ) = (τ, ∇ · ψ) − R[F , ψ],

(w, ∇· φ) = −(g, φ), for all ψ, φ ∈ C

∞

necessarily w = u

and τ = π

+ const a.e. in R

Proof. The proof of the existence of u

, π

satisfying (VII.5.3) is entirely anal-

ogous to that of Theorem VII. 4.1. The only point that deserves a littl e care is

to show tha t the constant c in (VII.5.3) can be taken independent of ε. To see

how this can be done, we shall sketch the proof of (VII.5.3) when F ∈ C

∞

)

and g ≡ 0, leaving to the reader the simple tasks of giving details and extend-

ing the result to the general case. As in Section VII.4, we ﬁrst ma ke a change

of variable of the type (VII. 4.3), which formally brings (VII.5.2) into a similar

system having R = 1 and εR in place of ε. L et us denote by (VII. 5.2

) this

latter system. We next construct a solution to (VII.5.2

) by Fourier transform,

that is,

(x) =

2π

ix·ξ

(ξ)dξ

(x) =

2π

ix·ξ

(ξ)dξ,

where

(ξ) =

−ξ

(ξ

+ εR + iξ

)

(ξ), m = 1, 2

(ξ) = i

(ξ)ξ

and

F is the Fourier transform of F . Setting

(ξ) =

− ξ

(ξ

+ εR + iξ

)

, (VII.5.4)

using the same reasonings as in the proo f of Lemma VII.4.2 one shows with

no diﬃculty that the functions

(ξ), ξ

(ξ), `, s, m, k = 1, 2,

satisfy the assumptio n of Lemma VII.4.1 with β = 2/3, β = 1/3 and β = 0,

respectively, with a constant M independent of εR. The same is true for

462 VII Steady Oseen Fl ow in Exterior Domains

(ξ), with β = 1/2 , and for ξ

(ξ), with β = 0, k = 1, 2. As a consequence,

arguing exactly as in the proof of Theorem VII.4.1, we obtain (VII.5.3) with

a constant c independent of ε. Let us next show

∈ W

1,q

). (VII.5.5)

By Lemma VII.4.1, it suﬃces to prove, for some C = C(ε, R) and all `, m, k =

1, 2

|ξ

| |ξ



∂

∂ξ



≤ C

|ξ

| |ξ



∂

(ξ

)

∂ξ



≤ C

with κ

, κ

= 0, 1 and κ

+ κ

= κ. We now have, wi th c = c(κ

, κ

|ξ

| |ξ



∂

∂ξ



≤ c

+ εR + |ξ

≤

εR

and

|ξ

| |ξ



∂

(ξ

)

∂ξ



≤ c

|ξ|

+ εR + |ξ

≤

(εR)

1/2

and (VII.5 .5) follows. Finally, let w, τ satisfy conditio ns (a) and (b) stated in

the lemma and set v = w −u

, p = τ −π

. We then obtain that v obeys the

identity

(∇v, ∇ϕ) −R(v,

∂ϕ

∂x

) + ε(v, ϕ) = 0, for all ϕ ∈ D(Ω). (VII. 5.6)

However, by embedding Theorem II.3.2 , it follows that u

∈ W

1,2

) and

since ∇ · v = 0 we conclude v ∈ H

). By continuity, we may then extend

identity (VII.5.6) to all ϕ ∈ H

) and, in particular, we may take ϕ = v

to deduce

(∇v, ∇v) + ε(v, v) = R(v,

∂v

∂x

). (VII.5.7)

Since

(v,

∂v

∂x

) = 0,

(VII.5.7 ) yields v = 0 a.e. in R

and, consequently, from (b) and the a nalog ous

identity written for u

, we have

(p, ∇ · ψ) = 0, for all ψ ∈ C

∞

implying τ = π

+ const a.e. in R

. The pro of of the lemma is therefore

complete. ut

VII.5 Existence of Plane Flow in Exterior Domains 463

The second step consists in proving the existence of a generalized solution

to (VII.5.1), that is, of a ﬁeld v : Ω → R

satisfying the requirements (i)-(iv)

of Deﬁnition VII.1.1 with q = 2 , along with the identity

(∇v, ∇ϕ) − R(

∂v

∂x

, ϕ) + ε(v, ϕ) = −R[f , ϕ], for all ϕ ∈ D(Ω). (VII.5.8)

We have

Lemma VII.5.2 Let Ω be a locally Lipschitz, exterior domain of R

, and l et

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2,2

(∂Ω).

Then, for all ε ∈ (0, 1] there exists a generalized solution v

to (VII.5.1). This

solution veriﬁes the estimate

εkv

+ kv

2,Ω

+ |v

1,2

≤ c



R|f|

1,2

+ (1 + R)kv

∗

1/2,2(∂Ω)



(VII.5.9 )

for all R > δ(Ω

) and with a constant c

= c

(R, Ω). Moreover, denoting by

∈ L

loc

(Ω) the corresponding pressure ﬁeld,

we have

2,Ω

/ R

≤ c

{R|f|

−1,2

+ (1 + R)|v

1,2

} (VII.5.10)

for a ll R > δ(Ω

) and with a constant c

= c

(R, Ω).

Proof. Again, the proof is compl etely analogous to that given in Theorem

VII.2.1. Actually, we look for a solution v

of the form

= w

+ V

+ σ,

where V

is a suitable extension of v

∗

constructed exactly as in the proof of

Theorem VII.2.1 . In addition,

σ = −

4π

∇



log

|x|



∂Ω

∗

· n

(the origin of coo rdinates having been taken in

Ω

) and w

∈ H

(Ω) veriﬁes

the identity

(∇w

, ∇ϕ) −R(

∂w

∂x

, ϕ) + ε(w

, ϕ)

= −R[f , ϕ] − (∇V

, ∇ϕ) − R(

∂V

∂x

∂σ

∂x

, ϕ),

for all ϕ ∈ D(Ω). Using the Galerkin method and exploiting the properties

of V

, σ, a nd a , we easily show the existence of w

obeying the previous

identity and the following inequality

Associated to v

in a way completely analogous to that used in Lemma VII.1.1

for generalized solutions with ε = 0.

464 VII Steady Oseen Fl ow in Exterior Domains

εkw

+ |w

1,2

≤ c



R|f|

−1,2

+ (1 + R)kv

∗

1/2,2(∂Ω)



(see the proof of Theorem VI I.2.1). From this we deduce the existence of a

generalized solution v

satisfying (VII.5.9). The estimate for p

is obtained

again as in the proof of Theorem VII.2.1. No tice that this time, si nce ε 6= 0,

we are abl e to control the behavior of v

at inﬁnity and to show, in particular,

the validity of condition (iv) of Deﬁnition VII.1. 1. Actually, for ε > 0, we have

∈ W

1,2

(Ω) and so, putting |x| = r and

I(r) =

2π

(r, θ)|

dθ, r > δ(Ω

we recover

I(r) ∈ L

(0, ∞),

∈ L

(0, ∞),

which implies I(r) = o(1) as r → ∞. ut

The third step is to prove that if, in addition to the assumptions made in

Lemma VII.5.2, f belongs to L

(Ω) for some q ∈ (1, 3/2), then v

and its

derivatives belong to suitable Lebesgue spaces and satisfy there an estimate

in terms of the data uniformly in ε ∈ (0, 1]. Speciﬁcally, we have

Lemma VII.5.3 Let Ω, f , and v

∗

satisfy the hypotheses of Lemma VII.5.2.

Suppose, further, f ∈ L

(Ω), 1 < q < 3/2. Then, the generalized solution v

determined in Lemma VII.5.2 and the corresponding pressure ﬁeld p

verify,

in addition, for all R > δ(Ω

)

∈ D

2,q

(Ω

) ∩ D

1,3q/(3−q)

(Ω

) ∩ L

3q/(3−2q)

(Ω)

∈ L

2q/(2−q)

(Ω) ∩ D

1,q

(Ω)

∂v

∂x

∈ L

(Ω)

∈ D

1,q

(Ω

)

along with the estimate

2q/(2−q)

+ |v

1,q



∂v

∂x



+bkv

3q/(3−2q)

+ R

1/3

1,3q/(3−q),Ω

R + |v

2,q ,Ω

R + |p

1,q,Ω

≤ c



R(kfk

+ (1 + R)|f |

−1,2

) + (1 + R)

∗

1/2,2(∂Ω)



(VII.5.1 1)

where b = min{1, R

2/3

} and c = c(q, Ω, R).

Proof. Let χ ∈ C

∞

) be zero in B

R/2

and one in B

, for some arbitrarily

ﬁxed R > δ(Ω

). Setting u

= χv

, π

= χp

it is easy to show that u

, π

satisfy (VII.5.2)

with

In the generalized sense.

VII.5 Existence of Plane Flow in Exterior Domains 465

RF = ∆χv

+ 2∇χ · ∇v

+ Rv

∂χ

∂x

+ p

∇χ + Rχf

g = ∇χ · v

By the properties of χ, we readily deduce the estimates

RkF k

≤c

[Rkfk

+ (1 + R)kv

1,2,Ω

+ kp

2,Ω

]

kgk

≤c

2,Ω

|g|

1,q

≤c

1,2,Ω

(VII.5.1 2)

with c

= c

(χ), i = 1, 2. Using these inequalities along with Lemma VII.5.2

we deduce tha t

F ∈ L

), g ∈ W

1,q

), 1 < q < 3/2

and that

RkF k

+ |g|

1,q

+ Rkgk

is increased through the right-hand side of (VII.5.11). From Lemma VI I.5.1

we then deduce

∈ W

2,q

) ∩ D

1,3q/(3−q)

) ∩L

3q/(3−2q)

)

∈ D

1,q

)

∈ L

2q/(2−q)

) ∩D

1,q

)

∂u

∂x

∈ L

)

and that u

, π

satisfy (VII.5.3). The pro of then follows from (VII.5.9) and

(VII.5.1 2), and by recalling that χ = 1 in B

and that

2q/(2−q),Ω

+ |v

1,q,Ω



∂v

∂x



q,Ω

+ kv

3q/(3−2q),Ω

≤ ckv

1,2,Ω

We are now in a position to prove the following.

Theorem VII.5.1 Let Ω be a two-dimensional, locally Lipschitz exterior

domain. Then, given

f ∈ D

−1,2

(Ω) ∩ L

(Ω), 1 < q < 3/2,

∗

∈ W

1/2,2

(∂Ω) ,

there exists a unique generalized solution v to (VII.0.2) and (VII.0.3). More-

over, for all R > δ(Ω

) this solution veriﬁes