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

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566 VIII Steady Generalized Oseen Flow in Exterior Domains

with C

= C

(q). Next, let f be merely in L

), and denote by {f

} ⊂

∞

) a sequence converging to f in L

). We shall show that the se-

quence of corresponding solutions {v

, p

} can be modiﬁed by adding to v

suitable linear function, in such a way that the new sequence converges to a

solution (v, p) to (VI II.7.17) satisfying the estimates stated in the lemma. We

begin by considering the case q ∈ (1 , ∞). From (VIII.7.2 9) and the uniqueness

Theorem VIII.7.1, it follows that {v

, p

} is Cauchy in

2,q

) ×

1,q

and, consequently, so is {

, p

}, where

:= v

−a

−b

−c

− d

:= (D

)

, b

:= (D

)

, c

:= (D

)

, d

:= v

(VIII.7.30)

Thus, by Lemma II.6.2 and (VIII.7.29), there exists at least one (

v, p) ∈

2,q

) ×D

1,q

) such that

→ D

v , ∇p

→ ∇p in L

) , (VIII.7.31)

which satisﬁes

2,q

+ |p|

1,q

≤ C

kfk

, (VIII.7.32)

In addition, also by means of the ﬁrst of the obvious properties

= 0 , for all R > 0 and i, j = 1, 2, 3 , i 6= j , (VIII.7.33)

we obtain

)

= (D

)

= (D

)

= 0 ,

and so, from a double application of Theorem II.5.4, Exercise II.6.1 and

(VIII.7.31), we obtain

→

v in W

1,q

), for all R > 0 . (VII I.7.34)

Thus, noticing that ∇ ·

= 0 for all k ∈ N, we deduce, in particular,

∇ ·

v = 0 . (VIII.7.35)

Since

×x ·∇x

= 0 , e

× x · ∇x

= −x

, e

× x · ∇x

= x

, (VIII.7.36 )

we have

×x ·∇v

= e

×x · ∇

+ x

−x

×v

= e

+ e

× (x

+ x

+ d

) .

Consequently, from (VIII. 7.17)

, which wi th f ≡ f

is satisﬁed by (v

, p

)

for a ll k ∈ N, we obtain

VIII.7 Existence, Uniqueness, and L

-Estimates. T he case R = 0 567



∆

+ T (e

× x · ∇

− e

), ψ



−(∇p

, ψ) − (f , ψ)

= T (x

+ x

+ e

×(x

+ d

), ψ) ,

: = −c

+ e

× b

, C

:= b

+ e

×c

(VIII.7.37)

for all ψ ∈ L

∞

) with bounded support. In view of (VIII.7.31) and

(VIII.7.34), it follows that as k → ∞,



∆

+ T (e

× x · ∇

− e

), ψ



− (∇p

, ψ) − (f , ψ)

→



∆

v + T (e

×x · ∇

v − e

v), ψ



−(∇p, ψ) − (f, ψ) ,

(VIII.7.38)

for all functions ψ speciﬁed above. Denote by χ

the characteristic function

of B

. Then, if we choose in (VIII.7.37), in order, ψ = χ

, ψ = χ

, i =

1, 2, 3, recall (VIII.7.33), and use (VIII.7.38), it is easily shown that there are

d, a, B, C ∈ R

such that

× d

→ e

×d , e

×a

→ e

× a , B

→ B , C

→ C , as k → ∞,

(VIII.7.39)

where the vectors a, B, C may be taken to satisfy

a · e

= 0 , B = B

+ e

× C , C = C

−e

× B . (VIII.7.40)

Combining (VIII.7.38)–(VIII.7.40), by the arbitrariness of ψ we thus deduce

the validity of the following equation, a.e. in R

∆

v+ T (e

× x · ∇

v − e

v) − ∇p − f

= T (x

+ e

×C) + x

−e

× B) + e

× (x

a + d)) .

(VIII.7.41)

Set

∗

v + x

a + d + x

−x

Observing that with the help of (VIII.7.36), we have

v + x

×a + e

× d = e

×(

v + x

a + d + x

− x

) ,

= e

×v

∗

×x ·∇

v − x

− x

= e

×x · ∇(

v − x

+ x

)

= e

×x ·∇(

v−x

+ x

a + d)

= e

×x · ∇v

∗

equation (VIII.7.41) can be rewritten as follows:

∆v

∗

+ T (e

×x ·∇v

∗

− e

×v

∗

) −∇p − f = T (x

× C − x

×B)) .

(VIII.7.42)

Moreover, by (VIII.7 .35) and (VIII.7.40)

, we obtain

568 VIII Steady Generalized Oseen Flow in Exterior Domains

∇ · v

∗

= 0 . (VIII.7.43)

We next observe that setting B

= (0, B

, B

) and C

= (0, C

, C

), from

(VIII.7.40)

2,3

, we deduce

= e

×C = e

× C

, C

= −e

× B = −e

× B

. (VIII.7.44)

Therefore, i f we let v := v

∗

−x

), equation (VIII.7.42) delivers

∆v + T (e

×x ·∇v

∗

−

+ x

) − e

×v) = ∇p + f . (VI II.7.45)

However, employing the identities (VIII.7.36)

2,3

, we have

×x·∇v

∗

−

) = e

×x·∇



∗

−x

)) = e

×x·∇v ,

so that from the la tter and (VIII.7.45), we establish that (v, p) satisﬁes

(VIII.7.17)

. Since v and

v diﬀer by a linear function of x, by (VIII.7.32), we

infer that (v, p) a lso satisﬁes (VIII.7.19). Finally, noting that by (VIII.7.44)

= B

, and using (VIII.7.43), we get ∇·v = 0, and we conclude that (v, p)

satisﬁes all the properties stated i n the lemma, which is thus proved for a

generic q ∈ (1, ∞). If q ∈ (1, 3), we deﬁne

:= v

− v

, and proceed

as in the previous proof, by formally taking a

= b

= c

= 0. The only

thing tha t we have to noti ce is that when q ∈ (1, 3), the ﬁelds v

and p

converge (strongly) also in

3q/(3−q)

), in view of the following inequali ties

(see Theorem II.6.3, and, in parti cular, (II. 6.49)):

3q/(3−q)

≤ c

2,q

3q/(3−q)

≤ c

1,q

, for all k ∈ N.

As a consequence, the ﬁelds

v (and therefore v :=

v + d; see (VIII.7.39)) and

p w ill satisfy the same inequality, which, together with (VIII. 7.32), completes

the proof also when q ∈ (1, 3). Fina lly, if q ∈ (1, 3/2), ag ain by Theorem II.6.3

and by (VIII.7.29) the approximating soluti on {v

, p

} satisﬁes the inequality

3q/(3−2q)

+ |v

3q/(3−q)

+ |v

2,q

+ |p

1,q

≤ c

, for all k ∈ N.

(VIII.7.46)

Thus, from (VIII.7.46 ) and the uniqueness Theorem VIII.7.1, it follows that

, p

} is Cauchy in the (Banach) space of functions having ﬁnite norm on

the left-hand side of (VII I .7.46). Therefore, the sequence converges to some

(v, p) that, on the one hand, satisﬁes (VIII .7.46) with f in place of f

, and, on

the other hand, as shown by a simple argument, satisﬁes (VIII.7.17). Fina lly,

assume f ∈ L

) for all q ∈ (1, ∞). By taking, in particular, q ∈ (1, 3/2),

from the existence result just shown, we obtain a solution (v, p) satisfying

(VIII.7.18)–(VIII.7.23) for this speciﬁc value of q. However, by the existence

result and the uniqueness Lemma VII I .7.1, (v, p) will satisfy (VIII.7.18)–

(VIII.7.23) for all q ∈ (1, 3/2). Next, let q ∈ [3/2, 3) and let

VIII.7 Existence, Uniqueness, and L

-Estimates. T he case R = 0 569

, p

) ∈ [D

2,q

) ∩ D

1,3q/(3−q)

)] × [D

1,q

) ∩ L

3q/(3−q)

)]

be the solution previously constructed. We claim that

= D

v ∇v

= ∇v , and p

= p , a.e. in R

. (VIII.7.47)

In fact, again by Lemma VIII.7.1, the ﬁrst relation in (VIII. 7.47) follows

immediately. This latter implies ∇v

= ∇v+A, where A is a constant second-

order tensor. However, since f or s ∈ (1, 3/2),



|∇v

3q/(3−q)

+ |∇v|

3s/(3−s)



< ∞,

there exists an unbounded sequence {ρ

} ⊂ (0, ∞) such tha t

lim

k→∞



|∇v

(ρ

, ω)|

3q/(3−q)

+ |∇v(ρ

, ω)|

3s/(3−s)



dω = 0,

which implies

lim

k→∞

(|∇v

(ρ

, ω)|+ |∇v(ρ

, ω)|) dω = 0,

and so we conclude that A = 0, which proves the second relation in

(VIII.7.47). As for the pressure, we observe that since both (v

, p

) and (v, p)

obey (VIII.7.17), by a simple argument we show that, setting π := p

− p, it

is (∇π, ∇ψ) = 0, for all ψ ∈ C

∞

). Thus, π is harmonic in R

, and since

∈ L

3q/(3−q)

), p ∈ L

3s/(3−s)

), s ∈ (1, 3/2), the third condi tion in

(VIII.7.47) fo llows from Exercise II.11.11. Fina lly, suppose q ∈ [3, ∞). Then,

by the existence part of the lemma, we ﬁnd a soluti on (v

, p

) such that

, p

) ∈ D

2,q

) × [D

1,q

) .

We claim

= D

v and ∇p

= ∇p , a.e. in R

. (VII I.7.48)

In fact, again by the uniqueness Lemma VIII.7.1, the ﬁrst condition in

(VIII.7.48) is at once veriﬁed. Concerning the second one, we can show, as

befo re, that p

−p is harmonic in R

and since p

∈ D

1,q

), p ∈ D

1,s

) for

s ∈ (1, 3/2), from Exercise II.11.11 we infer the validity of the second relation

in (VIII.7.48), and the proo f of the lemma is complete. ut

Exercise VIII.7.1 Let v be the velocity ﬁeld of the solution determined in Lemma

VIII.7.2 corresponding to f ∈ L

), q ∈ (1, ∞), and let (x

, r, θ) be a system of

cylindrical coordinates. Show that ∂v

/∂θ ∈ L

). Show also that this latter

implies (v

− v

) ∈ L

), where v

= v

, r) :=

2π

, r, θ) dθ. Hint: Use

(VIII.7.19) and the Wirtinger inequality (II.5.17).

570 VIII Steady Generalized Oseen Flow in Exterior Domains

The next result concerns an extension of the previous lemma to the case

of an exterior domain.

Theorem VIII.7.2 Let Ω be an exterior domain of class C

. Then, for any

given

f ∈ L

(Ω) , v

∗

∈ W

2−1/q,q

(∂Ω) , q ∈ (1, 3/2) , (VIII.7.49)

there exists at least one solution, (v, p), to the generalized Oseen problem

(VIII.0.7), (VIII.0 .2) with R = 0, such that

v ∈ D

2,q

(Ω) ∩ D

1,3q/(3−q)

(Ω ∩L

3q/(3−2q)

(Ω) , p ∈ D

1,q

(Ω ∩ L

3q/(3−q)

(Ω) .

(VIII.7.50)

Moreover, the following estimate holds:

kvk

3q/(3−2q)

+ |v|

3q/(3−q)

+ |v|

2,q

+ |p|

1,q

+ T ke

×x · ∇v − e

×vk

≤ C



kf k

+ kv

∗

2−2/q,q(∂Ω)



(VIII.7.51)

where C = C(q, Ω, B) whenever T ∈ [0, B]. Finally, suppose that for some

r ∈ (1, ∞) a nd all R > δ(Ω

), (v

, p

) ∈ W

2,r

(Ω

) × W

1,r

(Ω

) is another

solution to (VIII.0 .7) with R = 0 corresponding to the same data and that

v ∈ L

(Ω) , for some q ∈ (1 , ∞) . (VIII.7 .52)

Then v(x) = v

(x) , p(x) = p

(x) + p

for a .a. x ∈ Ω, and some constant p

Proof. We begin by observing that the uniqueness part is an immediate con-

sequence of Theorem VII I .7.1. We thus proceed to the proof o f existence. For

given f and v

∗

satisfying the assumptio ns of the theorem, let {f

} and {v

∗k

}

be sequences of smooth functions, with f

of bounded support for each k ∈ N,

converging to f and v

∗

in the L

(Ω)- and W

2−1/q,q

(∂Ω)-norms, respectively.

We then denote by v

, k ∈ N, the (unique) generalized solution corresponding

to f

, v

∗k

, and by p

∈ W

1,2

(Ω

) the associated pressure ﬁeld. We also set

= ψ (v

+ Φ

σ) + w

, φ

= ψ p

where ψ is the “cut-oﬀ” function introduced in the proof of Theorem VIII .7.1,

∂Ω

∗k

· n ,

σ is deﬁned in (VIII.1.5), and ﬁnally, w

satisﬁes (VIII.7.12) with v ≡

+ Φ

σ. Using the properties of σ, it is at once checked that (VIII.7.13) is

satisﬁed. We next observe that u

and φ

satisfy (VIII.7.17) with f ≡ F

where

See Remark VIII.7.1.

VIII.7 Existence, Uniqueness, and L

-Estimates. T he case R = 0 571

:= f

+ ∆w

+ T



×w

−e

× w



+ 2Φ

∇ψ ·∇σ + Φ

σ∆ψ

−p

∇ψ + 2∇ψ ·∇v

+ v

∆ψ ,

and where we have used (VIII.1.8). Therefore, recalli ng (VIII.7.12)

, that for

all R > δ(Ω

∈ W

2,2

(Ω

) ∩ L

(Ω) , p

∈ W

1,2

(Ω

) (VIII.7.53)

(see Remark VIII.1.1), and tha t moreover, by the trace Theorem II.4.1, |Φ

| ≤

ckv

1,r,Ω

2σ

, for any r ≥ 1, we obtain, after a simple computati on,

≤ C (kf

+ kv

1,q,Ω

2σ

+ kp

q,Ω

2σ

) , for all q ∈ (1, 3/2),

(VIII.7.54)

where C = C(σ, Ω, q, B). Now, aga in by (VI II.7. 53), we have

∈ L

) ∩ W

2,2

) , φ

∈ W

1,2

) ,

for all R > 0, and so by Lemma VIII.7.2, Lemma VIII.7.1 , and (VIII.7.54), we

obtain, on the one hand, that (u

, φ

) is in the class (VIII.7.50), for all k ∈ N,

and that, on the other hand, it must satisfy (VIII.7.51) with f replaced by

. Recalling that u

(x) = v

(x), φ

(x) = p

(x) for all x ∈ Ω

2σ

, and taking

into account (VIII.7.54 ), we thus deduce

3q/(3−2q),Ω

2σ + |v

3q/(3−q),Ω

2σ + |v

2,q ,Ω

2σ

+ |p

1,q ,Ω

2σ

≤ C (kf

+ kv

1,q,Ω

2σ

+ kp

q,Ω

2σ

) ,

(VIII.7.55)

with C = C(q, Ω, σ). However, by Lemma VIII.6.1, we also have

2,q,Ω

2σ

+kp

1,q,Ω

2σ

≤ c

(kf

q,Ω

3σ

+ kv

∗k

2−1/q,q,∂Ω

+ kv

q,Ω

3σ

+ kp

q,Ω

3σ

(VIII.7.56)

where c

= c

(q, σ, Ω, B). Therefore, combining (VIII.7.55) and (VIII.7.56)

we deduce

3q/(3−2q)

+|v

3q/(3−q)

+ |v

2,q

+ |p

1,q

≤ c



+ kv

∗k

2−1/q,q(∂Ω)

+ kv

1,q Ω

3σ

+ kp

q,Ω

3σ



(VIII.7.57)

where c

= c

(q, Ω, σ, B). The next task is to prove the exi stence of a constant

= c

(q, Ω, σ, B) such that

1,q,Ω

3σ

+ kp

q,Ω

3σ

≤ c



+ kv

∗k

2−1/q,q(∂Ω)



. (VIII.7.58)

The proof of (VIII.7.58) i s based on a contradiction argument that uses (i) the

embedding W

1,q

(Ω) ,→,→ L

(Ω

) (see Exercise II.5.8), and (ii) the unique-

ness of solutions to (VIII.0 .7), (VIII.0.2) with R = 0 and T ∈ [0, B], in

the class of functions (VIII.7.50). Since the argument is entirely analogous to

572 VIII Steady Generalized Oseen Flow in Exterior Domains

that employed in the proofs of Theorem V.5. 1, Theorem V.8 .1, and Theorem

VII.7.1, it will be omitted. From (VII I.7.57) and (VIII.7.58 ), we thus infer

3q/(3−2q)

+|v

3q/(3−q)

+|v

2,q

+|p

1,q

≤ c



+ kv

∗k

2−1/q,q(∂Ω)



(VIII.7.59)

with c

= c

(q, Ω, σ, B). We now l et k → ∞ in this relation and, again by a

by now standard reasoning, we show that {(v

, p

)} tends, in the topology

deﬁned by the left-hand side of (VIII.7.59), to a solution (v, p) to (VIII.0.7),

(VIII.0.2), with R = 0, satisfying all the statements of the theorem, which is

therefore completely proved. ut

Remark VI II.7.1 We would like to specify the way in which solutions con-

structed in the previous theorem satisfy the condi tion at inﬁnity (VIII.0.2).

Since q ∈ (1, 3/2), we have v ∈ D

1,s

(Ω), where s := 3q/(3 − q) < 3. Conse-

quently, taking also i nto account that v ∈ L

3q/(3−2q)

(Ω), from Lemma II.6.3

we have

|v(|x|, ω)|dω = o



|x|

2−3/q



as |x| → ∞.



Exercise VIII.7.2 Let the assumptions of Theorem VIII.7.2 on Ω, f , and v

∗

satisﬁed, and suppose, in addition, that f ∈ L

(Ω), v

∗

∈ W

2−1/s,s

(∂Ω), for some

s ∈ (1, ∞). Show that the corresponding solution (v, p), besides the properties stated

in t hat theorem, satisﬁes also the foll owing ones:

(v, p) ∈ D

2,s

(Ω) ×D

1,s

(Ω) ,

|v|

2,s

+ |p|

1,s

≤ C

kf k

+ kv

∗

2−1/s(∂Ω)

with C = C(q, Ω, B), w henever T ∈ [0, B]. Moreover, show that

lim

|x|→∞

v(x) = 0 if s ∈ (1, 3/2) ; lim

|x|→∞

∇v(x) = 0 if s ∈ (1, 3) .

Hint: Use the argument s of the proof of Theorem V.4.8.

VIII.8 Existence, Uniqueness, and L

-Esti mates. The

Case R 6= 0

We begin by proving existence of solutions, with corresponding estimates,

to the system (VIII.5.1)

1,2

, when f ∈ L

). To this end, we will use the

following result of Farwig (2006, Theorem 1.1) which we state wi tho ut proof;

see also Galdi & Kyed (2011b, Theorem 1.1) for a simple proof.

Lemma VIII.8.1 Let f ∈ L

), for some q ∈ (1, ∞), and R > 0. Suppose

that the problem

VIII.8 Existence, Uniqueness, and L

-Estimates. The Case R 6= 0 573

∆v + R

∂v

∂x

+ T (e

×x ·∇v − e

×v) = ∇p + f

∇· v = 0











a.e. in R

(VIII.8.1)

has a solution (v, p) ∈ D

2,s

) × D

1,s

), for some s ∈ (1, ∞). Then

∂v

∂x

, (e

×x ·∇v − e

×v) ∈ L

) ,

and the following estimate holds:



∂v

∂x



+ T ke

×x · ∇v − e

×vk

≤ C



1 +



kf k

We are now in a position to prove the following.

Lemma VIII.8.2 Let f ∈ L

), for some q ∈ (1, ∞), and R > 0. Then,

problem (VIII.8.1) has at least one solution (v, p) such that

(v, p) ∈ D

2,q

) × D

1,q

) ,

∂v

∂x

, (e

×x · ∇v − e

×v) ∈ L

) ,

(VIII.8.2)

and that satisﬁes the inequalities

|v|

2,q

+ |p|

1,q

≤ C

kf k

(VIII.8.3)

and



∂v

∂x



+ T ke

× x · ∇v − e

× vk

≤ C



1 +



kf k

(VIII.8.4)

with C

= C

(q). If q ∈ (1, 4), then there is a solution that, in additi on to

(VIII.8.2)–(VIII.8.4), satisﬁes also

v ∈ D

1,4q/(4−q)

) ,

1/4

|v|

1,4q/(4−q)

≤ C



1 +



kfk

(VIII.8.5)

with C

= C

(q). Furthermore, if q ∈ (1, 2), then we can ﬁnd a solution tha t,

in addition to (VIII.8.2)–(VIII.8.5), satisﬁes f urther,

v ∈ L

2q/(2−q)

) , p ∈ L

3q/(3−q)

) (VIII.8.6)

along with the inequalities

In fact, this summability property for p bel ow requires only q ∈ (1, 3).

574 VIII Steady Generalized Oseen Flow in Exterior Domains

1/2

kvk

2q/(2−q)

≤ C



1 +



kf k

kpk

3q/(3−q)

≤ C

kf k

(VIII.8.7)

where C

= C

(q).

Finally, if f ∈ L

) fo r all q ∈ (1, ∞), then there exists a solution (v, p)

to (VIII.8.1) satisfying (VIII.8 .2)–(VIII.8 .7) for all speciﬁed values of q.

Proof. The proof of this result follows the same main arguments used in that of

Lemma VIII.7.2, and therefore will be only sketched here leaving to the reader

the (straightforward) task of ﬁlling in the details. We ta ke ﬁrst f ∈ C

∞

Thus, from Theorem VIII.5.1, we know that problem (VIII.8.1) has one and

only one smooth solution (v, p) such that

[]v[]

1,R

+ []∇v[]

2,R

+ []D

v[]

2,R

< ∞,

|p|

1,2

+ kpk

< ∞, all r ∈ (3/2, ∞) .

It is a simple exercise to show tha t

v ∈ L

) ∩ D

1,s

) ∩ D

2,s

) , all s

> 2, s

> 4/3, and s

> 1 ;

(VIII.8.8)

see also (VII.3.28 ) and (VI I .3.33). Therefore, arguing exactly as in the part

of the proof of Lemma VIII.7.2 leading to (VIII.7.29 ), we can show that v, p

satisfy the estimate (VIII. 8.3). From Lemma VI II.8.1 we then infer (VIII. 8.2)

together with the validity of (VIII.8.4). Take now f just in L

), and let

} ⊂ C

∞

) be a sequence converging to f in L

), and {(v

, p

)}

the sequence of corresponding solutions. From what we have shown, (v

, p

)

satisﬁes (VIII.8 .2)–(VIII.8.4), with f ≡ f

, for all k ∈ N and all q ∈ (1, ∞).

We then consider the ﬁelds

deﬁned in (VIII.7.30) by form ally setting a

0, and follow, step by step, the analogous proof given in Lemma VIII.7.2 for

the case of a generic q ∈ (1, ∞), to prove the existence of a solution for such

a value of q. T his solution satisﬁes the estimates (VIII.8.3) a nd that for D

in (VIII.8.4). Thus, from (VIII.8.1), we deduce that it satisﬁes (VIII.8.4).

The lemma is then proved for q ∈ (1, ∞). If q ∈ (1, 4), we forma lly rewrite

the generalized Oseen system (VIII. 8.1) for (v

, p

) as the following Oseen

system:

∆v

+ R

∂v

∂x

= ∇p

+ F

∇ · v

= 0











in R

, (VIII.8.9 )

where

:= f

−T (e

× x · ∇v

− e

×v

) .

From Theorem VII.4. 1 and the uniqueness property of the pair (v

, p

) we

ﬁnd that the sequence {v

} is Cauchy (also) in D

4q/(4−q)

), and thus, af-

ter modifying the v

by a constant vector and proceeding as in the proof

VIII.8 Existence, Uniqueness, and L

-Estimates. The Case R 6= 0 575

of Lemma VIII.7.2, we show that the modiﬁed sequence, together with the

corresponding sequence {p

}, converges to a solution to (VIII.8.1) with the

properties (VII I.8.2)–(VIII.8.5). In the last case, q ∈ (1, 2), again from The-

orem VII.4.1 applied to (VIII.8 .9), we obtain that the sequence {(v

, p

)}

satisﬁes (VII I .8.2)–(VI II.8.7) w ith f ≡ f

. Using agai n the uniqueness prop-

erty of (v

, p

) along with the estimates in (VIII.8 .3)–(VIII.8.7), it is routine

to show that the sequence {v

, p

)} converges to a solution (v, p) to (VIII.8.1)

satisfying the desired properties. Finally, assume f ∈ L

) for all q ∈ (1, ∞).

Then, we may follow verbatim the argument adopted in the proof of Lemma

VIII.7.2 to establish the existence of a solution satisfying the stated properties.

Details are agai n left to the reader.

With Lemma VIII.8.2 in hand, we shall now proceed to prove the main

result of this section. To this end, it is convenient to introduce a suitable

function class. Let Ω be an exterior domain of R

, and let v, p be vector and

scalar ﬁelds, respectively, deﬁned on Ω. We shall say that (v, p) is in the class

(Ω), q ∈ (1, 2), if

v ∈ D

2,q

(Ω) ∩ D

1,4q/(4−q)

(Ω) ∩ L

2q/(2−q)

(Ω) ,

∂v

∂x

, (e

× x · ∇v − e

× v) ∈ L

(Ω) ,

p ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω) .

Remark VI II.8.1 We o bserve that by Lemma II.6.1, C

(Ω) ⊂ W

2,q

(Ω

for a ll R > δ(Ω

), if Ω is locally Lipschitz. 

Our main result thus reads as follows.

Theorem VIII.8.1 Let Ω be an exterior domain of R

of cla ss C

, and

assume R > 0. Then, for any given

f ∈ L

(Ω) , v

∗

∈ W

2−1/q,q

(∂Ω) , q ∈ (1, 2) , (VIII.8.10)

there exists at least one corresponding solution v, p to the generalized Os-

een problem (VIII. 0.7), (VIII.0.2) in the class C

(Ω). Moreover, the following

estimate holds:

|v|

2,q



∂v

∂x



+ T |e

×x · ∇v − e

×vk

+ R

1/4

|v|

1,4q/(4−q)

1/2

kvk

2q/(2−q)

+ |p|

1,q

+ kpk

3q/(3−q)

≤ C



kf k

+ kv

∗

2−1/q,q(∂Ω)



(VIII.8.11)

with C = C(q, Ω, R, T). However, if q ∈ (1 , 3/2), we may take C =

C(q, Ω, B, T ) whenever R ∈ (0, B].