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

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326 V Steady Stokes Flow in Exterior Domains

it follows that

v ∈

2,q

(Ω), for q ≥ n/2.

The uniqueness of the solution is a consequence of Theorem V.3.4.

To prove (ii), we begin to make a suitable solenoidal extension of the ﬁeld

≡ A · x. L et w denote a solution to the problem

∇ · w = ∇ϕ · V

≡ g in Ω

w = 0 at ∂Ω

(V.4.19)

where ϕ is the “cut-oﬀ” function used in the proof of Lemma V.4.2 and Ω

a locally Lipschitz subdomain of Ω that contains the support of ϕ. Since

Ω

g = 0, g ∈ C

∞

(Ω

by Theorem III.3.3 we can take w ∈ C

∞

(Ω). Setting

a(x) = (1 − ϕ)V

− w, (V.4.20)

by (V.4.19)

a is solenoidal, belongs to C

∞

(Ω), vanishes near ∂Ω, and equals

at l arge distances. Since

a ∈ C

∞

(Ω), (V.4. 21)

we may use the same procedure adopted in the proof o f Theorem V.2.1 to show

the existence of a g eneralized solution v to (V.4.1)

such that v = u +a, with

u ∈ D

1,2

(Ω). Employing Theorem IV.4.3 and Theorem IV.5. 1 we deduce, as

befo re, that v and the corresponding pressure p are of class C

∞

(Ω) and satisfy

(V.4.17). Using the asymptotic expansion (V.3.1 7), (V.3.19), and (V.3.21) for

u and recalling (V.4.21 ), we deduce

v ∈ L

(Ω), for all q > 1.

Since v does not belong to any space D

1,r

(Ω) nor to any L

(Ω), from (V.4 .16)

we conclude

v ∈

2,q

(Ω), for q ≥ n,

which completes the proof of (ii).

Now, let h

, π

, i = 1, . . . , n be the solutions to (V.4 .1)

of type (i) cor-

responding to the n orthonormal vectors v

∞i

= e

. Likewise, let u

, τ

the n

− 1 sol utions to (V.4.1)

of the type (ii) correspondi ng to the n

− 1

matrices of zero trace E

, where

(

⊗ e

if i 6= j

⊗ e

−e

⊗e

if i = j 6= n.

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 327

It is readily seen that the system constituted by the n

+n−1 vectors {h

, u

}

is linearly independent. Actually, assume per absurdum that there are noniden-

ticall y zero constants α

, α

∈ R such that

(x) +

(x) = 0 for all x ∈ Ω,

where the prime means that the term i = j = n i s excluded from the summa-

tion. From (V.3 .17)

, (V.3.19)

, and (V.3.21)

we would then obta in for all

suﬃciently large |x|

·x = O(1/|x|

n−2

which implies

= 0;

that is,

= α

= 0, for a ll i, j,

leading to a contradiction. Now, if v, p is a sol ution to (V.4.1)

with v ∈

2,q

(Ω), for som e q > 1, from Theorem V.3.3 we deduce the existence of

∞

∈ R

and of a traceless matrix B such that as |x| → ∞

v(x) = v

∞

+ B · x + O(1/|x|

n−2

). (V.4.22)

Clearly, by (V.4.16), we must have

(a) v

∞

= B = 0, if 1 < q < n/2 ;

(b) B = 0 , if n/2 ≤ q < n.

In case (a), by Theorem V.3.5, we have v ≡ 0 and so d = 0. In case (b)

we may write v

∞

= v

, for some v

∈ R, i = 1, . . ., n. Therefore

w ≡ v − v

, z ≡ p −v

is a solution to (V.4.1)

with w = o(1) as |x| → ∞ and so, again by Theorem

V.3. 5, we deduce w ≡ 0, which shows d = n if n/2 ≤ q < n. Finally, if q ≥ n,

we may write B = B

and thus, setting

w ≡ v −v

−B

, z ≡ p − v

− B

we again derive that w and z solve (V.4.1)

with w = o(1) as |x| → ∞,

which yields w ≡ 0, namely, d = n

+ n −1. The proof of the theorem is then

accomplished if n > 2. Let us consider the case where n = 2. We begin to

show the existence of two independent sol utions h

, π

, i = 1, 2, to (V.4.1)

with

∈

2,q

(Ω), 1 ≡ n/2 < q < n ≡ 2.

To this end, set

= U · e

, s

= −q · e

328 V Steady Stokes Flow in Exterior Domains

where U , q is the two-dim ensiona l Stokes fundamental solution. We look for

solutions of the form

= u

+ v

, π

= s

+ p

i = 1, 2,

where

∆v

= ∇p

∇· v

= 0

)

in Ω

= −u

at ∂Ω.

(V.4.23)

By Exercise III.3.5 we may extend −u

at the boundary to a solenoidal f unc-

tion v

∈ W

1,2

(Ω) of compact support in Ω. We then use the technique of

Theorem V.2.1 to show the existence of a weak solution v

to (V.4.23) of the

form

= w

+ v

, w

∈ D

1,2

(Ω)

with

∈ L

(Ω).

Actually, such a solution is of class C

∞

(Ω), by virtue of Theorem IV.4.3. It

is easy to prove that

∂B

T ( v

, p

) · n = 0 (V.4.24)

for a ll r > δ( Ω

). Actually, writing (V.4.23)

in the form

∇ · T (v

, p

) = 0,

we have

∂B

T (v

, p

) · n =

∂B

T ( v

, p

) · n =

∂B

(2D(w

) · n − p

n) (V.4.25)

for a ll R > r. By the summability properties of w

, p

we easily establi sh the

existence of a sequence {R

} ⊂ R

with

lim

k→∞

= ∞,

along which the right-hand side of (V.4.25) vanishes, which in turn implies

(V.4.24). From (V.3.17)–(V.3.21) we then obtai n for large enough |x| that

(x), p

(x) adm it the following representation:

(x) = v

+ O(1/|x| )

(x) = O(1/|x|

)

(V.4.26)

for som e constants v

and so

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 329

(x) = v

+ U (x) · e

+ O(1/|x|)

(x) = −q(x) · e

+ O(1/|x|

(V.4.27)

The solutions h

, π

, i = 1, 2, are linearly independent and, f urther,

∈

2,q

(Ω), q > n/2 (≡ 1). (V.4.28)

In fact, if

(x) + α

(x) = 0, for all x ∈ Ω,

from (V.4.27)

we would obtain for some v

∈ R

(α

+ α

) · U (x) = v

+ O(1/|x|),

which can b e attained if and only if v

= α

= 0. Moreover, by (V.4.27)

and the regularity properties of h

near the boundary (see (V.4.17)),

∈ D

1,r

(Ω) ∩ D

2,q

(Ω), r > n/(n − 1) ≡ 2, q > 1

and since h

6∈ L

(Ω) for any s ∈ (1, ∞) we obtain (V.4.28). As in the case

where n > 2, we shall next construct n

−1 (≡ 3) independent solutions u

with

∈

2,q

(Ω), q ≥ n (≡ 2).

Speciﬁcally, we look for solutio ns to (V.4.1)

of the form

= a

+ w

where a

are solenoidal extensions of type (V.4.20) of the ﬁelds V

0ij

= E

·x,

while w

solve the problem

∆w

= ∇τ

−∆a

∇ · w

= 0

)

in Ω

= 0 at ∂Ω.

Since a

satisﬁes (V.4.21), we apply the technique of Theorem V.2.1 to deduce

the existence of

∈ D

1,2

(Ω), τ

∈ L

(Ω).

As before, by Theorem V.3.2, for all suﬃciently large |x| we have

(x) = w

∞ij

+ O(1/|x|)

(x) = O(1/|x|

)

for som e constants w

∞ij

, and so

(x) = w

∞ij

+ E

· x + O(1/|x|

)

(x) = O(1/|x|

(V.4.29)

330 V Steady Stokes Flow in Exterior Domains

As in the case where n > 2 one shows that u

∈

2,q

(Ω), for all q ≥ n ≡ 2

and that the ﬁve vectors {h

, u

} form a linear independent system. Now, if

v, p is a solution to (V.4.1)

with v ∈

2,q

(Ω), for some q > 1, from Theorem

V.3. 3 we obtain that for large |x|, v(x) satisﬁes (V.4.22), with B = 0 if

1 < q < 2. Reasoning exactly as in the case where n > 2 one shows d = n ≡ 2

if q < n ≡ 2 and d = n + n

− 1 ≡ 5 if q ≥ n ≡ 2, thus com pleting the proo f

of the lemma. ut

With the aid of Lemma V.4 .3 and Lemma V.4.4 we can now obtain in

the case of exterior domains a result analogous to that proved, for bounded

domains, in the ﬁrst part of Theorem IV.6.1. To this end, for ﬁxed R > δ(Ω

)

and ` ≥ 0, ν ≥ 1 we set

kuk

ν,R;`,q

≡ kuk

ν−1,q,Ω

`+ν

i=1

|u|

i,q,Ω

The following lemm a holds.

Lemma V.4.5 Let Ω, f , v

∗

satisfy the same assumptio ns of Lemma V.4.3

and let v ∈

2,q

(Ω) be a solution to (V.4.1) corresponding to f and v

∗

. Then

v ∈ D

k+2,q

(Ω), p ∈ D

k+1,q

(Ω) for all k = 0, 1 , . . ., m and if q ≥ n we have

inf

(h,π)∈Σ

{kv − hk

2,R;m,q

+ kp − πk

1,R;m,q

}

≤ c



kf k

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



;

(V.4.30)

if n/2 ≤ q < n:

inf

(h,π)∈Σ

{|v −h|

1,r

+ kp − πk

+ kv − hk

2,R;m,q

+ kp − πk

2,R;m,q

}

≤ c



kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



(V.4.31)

where r = nq/(n −q); and if 1 < q < n/2:

kvk

+ |v|

1,r

+ kpk

+ kvk

2,R;m,q

+ kpk

1,R;m,q

}

≤ c



kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



(V.4.32)

where s = nq/(n − 2q).

Proof. In view of Lemma V.4 .3, we have to show only the va lidity of (V.4.30)–

(V.4.32). Consider ﬁrst the case where n < q. Taking into account that (h, π)

solves the homogeneous system (V.4.1)

, from (V.4.10) we derive

inf

(h,π)∈Σ

{kv− hk

2,R;m,q

+ kp − πk

1,R;m,q

}

≤ c

kf k

m,q

+ kv

∗

m+2−1/q,q(∂Ω)

+ inf

(h,π)∈Σ

{|v − h|

q,Ω

+ kp − πk

q,Ω

}

(V.4.33)

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 331

We claim the existence of a constant c

independent of v, p, f, and v

∗

such

that

inf

(h,π)∈Σ

{|v − h|

q,Ω

+ kp − πk

q,Ω

} ≤ c

(kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)

(V.4.34)

Actually, if (V.4.34) were not true, we could select two sequences {f

} ⊂

m,q

(Ω), {v

∗s

} ⊂ W

m+2−1/q,q

(∂Ω) with

→ 0 in W

m,q

(Ω)

∗s

→ 0 in W

m+2−1/q,q

(∂Ω)

(V.4.35)

as s → ∞, while the corresponding solutions {v

, p

} satisfy

inf

(h,π)∈Σ

{|v

− h|

q,Ω

+ kp

−πk

q,Ω

} = 1 fo r all s ∈ N. (V.4.36)

On the other hand, (V.4.35), (V.4.36), and (V.4.33) imply

inf

(h,π)∈Σ

{|v

− h|

2,R;m,q

+ kp

− πk

2,R;m,q

} ≤ M (V.4.37)

with M a constant independent of s. By the property of the inﬁmum, inequal-

ity (V.4.3 7) furnishes the existence of a sequence of solutions {v

≡ v

−h

≡ p

− π

} for some (h

, π

) ∈ Σ

such that

1,q,Ω

+ kD

q,Ω

+ kp

q,Ω

+ k∇p

q,Ω

≤ 2M. (V.4.38)

By the weak compactness of the space W

m,q

(Ω), 1 < q < ∞, and by the

compactness results of Exercise II.5.8, we deduce the existence of a sub-

sequence {v

, p

} and two pairs (v

(1)

, p

(1)

) ∈ W

1,q

(Ω

) × L

(Ω

) a nd

(V , P ) ∈ L

(Ω) × L

(Ω) such that

→ v

(1)

, p

→ p

(1)

weakly in W

1,q

(Ω

), strongly in L

(Ω

→ V , ∇p

→ P in L

(Ω).

(V.4.39)

By the deﬁniti on of weak derivative, it readily follows that D

(1)

and ∇p

(1)

exist in Ω

and that V = D

(1)

, P = ∇p

(1)

in Ω

. Fix now R

>R. In

Exercise V.4.7 the following inequality can be proved

kuk

q,Ω

≤ c



k∇uk

q,Ω

+ kuk

q,Ω



for all R

> R,

where c

= c

(Ω

, Ω

, q), and therefore from (V.4.38) we deduce

1,q,Ω

+ kp

q,Ω

≤ M

Thus, from {v

, p

} we can select a subsequence {v

, p

} such that

→ v

(2)

, p

→ p

(2)

weakly in W

1,q

(Ω

), strongly in L

(Ω

332 V Steady Stokes Flow in Exterior Domains

where

(2)

, p

(2)

) ∈ W

1,q

(Ω

) ×L

(Ω

Clearly,

(2)

= v

(1)

and p

(2)

= p

(1)

in Ω

V = D

(2)

, P = ∇p

(1)

in Ω

Iterating this procedure along a denumerable number of strictly increas-

ing dom ains of the type Ω

, m ∈ N, invading Ω, and using the classi-

cal diago nali zation method, we can eventually deﬁne a pair v, p in Ω with

v, p ∈ W

1,q

(Ω

), for a ll ρ > δ(Ω

) and, moreover, D

v, ∇p ∈ L

(Ω). It is

simple to check that v, p solve the homogeneous Stokes system and since, by

(V.4.16), D

2,q

(Ω) =

2,q

(Ω) for q ≥ n, by Lemma V.4.4 we must have

v = h, p = π, for some (h, π) ∈ Σ

. (V.4.40)

As a consequence, by (V.4.39)

and (V.4 .40), i t follows that

lim sup

→∞

( inf

(h,π)∈Σ

{ kv

− hk

q,Ω

+ kp

−πk

q,Ω

})

≤ lim

→∞



− h

−hk

q,Ω

+ kp

− π

−πk

q,Ω



= 0,

which contradicts (V.4.36). Thus (V.4.34) holds and the lemma follows when

q ≥ n. If n/2 ≤ q < n, we know from Theorem II.7 .4 and Theorem II.6.1 that

v obeys the inequality

|v|

1,r

≤ c

, (V.4.41)

where r = nq/(n − q). Likewise, by p ossibly adding a suitable constant to p,

we have

kpk

≤ c

|p|

1,q

≤ c

Therefore, in such a case, (V.4.33) can be strengthened by including in the

curly brackets on its left-hand side the quantity

|v − h|

1,r

+ kp − πk

Repeati ng the procedure adopted for the case where q ≥ n, we obtain this

time that the limit function v also belongs to D

1,r

(Ω) implying, in view of the

characterization given in (V.4.16), v ∈

2,q

(Ω). Also, v = h, p = π for some

(h, p) ∈ and so, reasoning as before, we then prove (V.4.34) and, consequently,

(V.4.31). Finally, if 1 < q < n/2, in conjunction with (V.4.41), from Theorem

II.6.1 we establi sh the validity o f the inequality

kvk

≤ c

|v|

1,r

≤ c

for s = nq/( n −2q). Then the limit function v belongs to L

(Ω) ∩ D

1,r

(Ω) ∩

2,q

(Ω) and so, by characterization (V.4.16), v ∈

2,q

(Ω). Again reasoning

as before, we show (V.4.34) and arrive at (V.4.32). The proof of the lemma is

complete. ut

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 333

Exercise V.4.7 Let Ω be an exterior, local ly Lipschitz domain of R

, n ≥ 2 and

let u ∈ L

(Ω

), ∇u ∈ L

(Ω

), R

> R > δ(Ω

). Use a contradiction argument

based on compactness to show the inequality

kuk

q,Ω

≤ c(k∇uk

q,Ω

+ kuk

q,Ω

where c = c(Ω

, Ω

, q).

Concerning the b ehavior at large distances of a solution v ∈

2,q

(Ω), we

have the fol lowing result.

Lemma V.4.6 Let Ω be an exterior domain in R

, and let v be a solution to

(V.4.1)

1,2

corresponding to f ∈ L

(Ω), with v ∈

2,q

(Ω). Then, if 1 < q < n

and t > n we have

lim

|x|→∞

∇v(x) = 0 (V.4.42)

uniform ly, while, if 1 < q < n/2 and t > n/2

lim

|x|→∞

v(x) = 0 (V.4.43)

uniform ly.

Proof. From Lemma V.3.1 we have the following representation for v:

(x) =

(x)

(d)

(x − y)f

(y)dy −

β(x)

(d)

(x − y)v

(y)dy (V.4.44)

with β(x) = B

(x) −B

d/2

(x). By (V.4.16)

we derive that, if 1 < q < n, then

∈ L

(Ω), r = nq/(n − q), and so diﬀerentiating (V.4.44) and recalling

the properties of U

(d)

and H

(d)

we deduce

(x)| ≤ c



k|x −y|

−n+1

(x)

kfk

t,B

(x)

+ k∇vk

r,B

(x)



Since t

< n/(n − 1), it follows that

k|x − y|

−n+1

≤ c

and so the preceding inequality implies (V.4.42). If 1 < q < n/2, from (V.4 .16)

we derive v ∈ L

(Ω), s = nq/(n − 2q) and from (V.4.44) we deduce

(x)| ≤ c



k|x −y|

−n+2

(x)

kf k

t,B

+ kvk

s,B

(x)



which shows (V.4.43). ut

We shal l next prove some existence results in the class of velocity ﬁelds

belonging to

2,q

(Ω).

334 V Steady Stokes Flow in Exterior Domains

Theorem V.4.6 Let Ω be an exterior domain of class C

m+2

, m ≥ 0. Given

f ∈ W

m,q

(Ω), v

∗

∈ W

m+2−1/q,q

(∂Ω), 1 < q < ∞, there exi sts a unique

solution to (V.4.1) such that

v, p ∈

2,q

(Ω) × D

1,q

(Ω) / Σ

Moreover,

v ∈

k=0

k+2,q

(Ω), p ∈

k=0

k+1,q

(Ω)

and estimates (V.4.30)–(V.4.32) are satisﬁed.

Proof. We approximate f and v

∗

by functions {f

} ⊂ C

∞

(Ω), {v

∗k

} ⊂

m+2−1/r,r

(∂Ω) any r ∈ (1, ∞), respectively. From Theorem V.2.1, for all s ∈

N there exists a g eneralized solution v

, p

∈ D

1,2

(Ω) × L

(Ω) corresponding

to f

, v

∗s

, and tending to 0 as |x| → ∞ in the case where n > 2. (If n = 2

this limit is undetermined.) Using Theorem IV.4.1 and Theorem IV.6.1 one

readily establishes (as in the proof shown in Remark V.4.6) that (at least)

∈ W

2,q

(Ω

), p

∈ W

1,q

(Ω

) for all R > δ(Ω

). From this information and

Theorem V.3.2 it follows that if n > 2

∈ L

(Ω) t > n/(n − 2),

∇v

∈ L

(Ω) r > n/(n −1),

∈ L

(Ω) q > 1,

while, if n = 2,

∇v

∈ L

(Ω) r ≥ n/(n −1), D

∈ L

(Ω) q > 1,

so that from (V.4.16) we obtain v ∈

2,q

(Ω) for all q > 1. (Notice that the

case 1 < q < n/2 is excluded if n = 2). The solutions (v

, p

) will then satisfy

(V.4.30)–(V.4.32), depending on the values of q and n. Assume q ≥ n. Given

ε > 0 from (V.4.30) and from the linearity of problem (V.4.1), for s

, s

suﬃciently large , we deduce

inf

(h,π)∈Σ

{|v

− v

− h|

2,q

+ |p

− p

−π|

1,q

} < ε. (V.4.45)

This relation implies that (v

, p

) is a Cauchy sequence in the quotient space

2,q

(Ω) × D

1,q

(Ω) / Σ

and so, by a cla ssical result of functional analysis,

there is an element (v, p) ∈

2,q

(Ω) × D

1,q

(Ω) to which v

, p

tend in the

quotient norm deﬁned by the left hand side of (V.4.45). Consequently, in view

of Lemma V.4.5, the theorem follows if q ≥ n. Likewise, if n/2 ≤ q < n, from

(V.4.31) we deduce

See, e.g., Schechter (1971, Chapter III, Theorem 5.3).

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 335

inf

(h,π)∈Σ

{|v

−v

−h|

1,r

+ |v

−v

−h|

2,q

+ |p

− p

− π|

1,q

} < ε,

namely, (v

, p

) is a Cauchy sequence in

2,q

(Ω)×D

1,q

(Ω) / Σ

and we prove

the theorem as before. Finally, if 1 < q < n/2, from Lemma V.4.4 it is Σ

= ∅

and v

− v

, p

− p

satisfy (V.4.25)

, thus b eing a Cauchy sequence in

2,q

(Ω) × D

1,q

(Ω) and the result again follows. The proof of the theorem is

therefore completed. ut

Let us now consider some consequences of Theorem V.4.6. Taking into

account the property of the inﬁmum, we immediately obtain

Theorem V.4.7 Assume Ω, f, and v

∗

satisfy the assumptio ns of Theorem

V.4. 6. Then there exists a solution v, p to (V.4.1) obeying the estimate

kvk

1,q,Ω

+ kpk

q,Ω

k=0

(| v|

k+2,q

+ |p|

k+1,q

)

≤ c



kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



where c = c(n, q, m, Ω, R).

We also have

Theorem V.4.8 Suppose Ω, f , and v

∗

satisfy the assumptions of Theorem

V.4. 6. Assume, in addition, f ∈ L

(Ω), v

∗

∈ W

2−1/t,t

(∂Ω), for some 1 < t <

n/2. Then, there exi sts one and only one solution v, p to (V.4.1) such that

v ∈

2,t

(Ω)∩



∩

k=0

k+2,q

(Ω)



p ∈ D

1,t

(Ω)∩



∩

k=0

k+1,q

(Ω)



∩ L

nt/(n−t)

(Ω).

Furthermore, v and p obey the following estimate

kvk

+ |v|

1,r

+ |v|

2,t

+ kpk

+ |p|

1,t

k=0

(|v|

k+2,q

+ |p|

k+1,q

)

≤ c



kf k

+ kfk

m,q

+ kv

∗

2−1/t,t(∂Ω)

+ kv

∗

m+2−1/q,q(∂Ω)



(V.4.46)

with r = nt/(n −t), s = nt/(n −2t), and c = c(n, q, t, m, Ω). Finally, we have,

as |x| → ∞,

|v(x)| = o (1/|x|

n/r−1

)

|∇v(x)| = o (1/|x|

n/t−1

)

|p(x)| = o (1/|x|

n/t−1

)

(V.4.47)