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

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336 V Steady Stokes Flow in Exterior Domains

and, if q > n,

lim

|x|→∞

v(x) = lim

|x|→∞

∇v(x) = 0 (V.4.48)

uniform ly.

Proof. From Theorem V.4.6 we deduce the existence of a solution to (V.4.1)

such that

v ∈

2,t

(Ω), p ∈ D

1,t

(Ω),

and verifying

kvk

+ |v|

1,r

+ |v|

2,t

+ kpk

+ |p|

1,t

≤ c(kf k

+ kv

∗

2−1/t,t(∂Ω)

). (V.4.49)

However, since f ∈ W

m,q

(Ω), v

∗

∈ W

m−2+1/q,q

(∂Ω), from Lemm a V.4.2 and

Lemma V.4.3 it foll ows that v ∈ D

k+2,q

(Ω), p ∈ D

k+1,q

(∂Ω), k = 0, . . . , m,

and that, moreover,

k=0

(|v|

k+2,q

+ |p|

k+1,q

) ≤ c



kf k

m,q

+ kv

∗

m+2−1/q,q(∂Ω)

+ kvk

q,Ω

+ kpk

q,Ω

) .

(V.4.50)

Given ε > 0, one can prove the f ollowing inequality (see Exercise V.4.8)

kuk

κ,Ω

≤ ckuk

σ,Ω

+ ε|u|

1,κ,Ω

, (V.4. 51)

for a ll κ, σ > 1, w ith c = c( ε, κ, σ, Ω

). Using (V.4.51) we obtain

kpk

q,Ω

≤ ckpk

+ ε|p|

1,q,Ω

, (V.4.52)

while using it twice furnishes

kvk

q,Ω

≤ c

kvk

+ c

|v|

1,r

+ εkD

q,Ω

. (V.4.53)

Using (V.4. 52) and (V.4.53) on the right-hand side of (V.4.50) and employing

(V.4.49) allows us to recover the estimate (V.4.46 ). Furthermore, relations

(V.4.47) follow from Lemma II.6.3, while (V.4.48) is a consequence of Lemma

V.4. 6. Finally, uniqueness is easily implied by Theorem V.3.4. The theorem

is, therefore, completely proved. ut

Exercise V.4.8 U se a contradiction argument based on the compactness results of

Exercise II.5.8 to show the validity of inequality (V.4.51).

Exercise V.4.9 The results proved in this section continue to hold if, more gen-

erally, ∇ · v = g 6≡ 0. In particular, show the validity of Theorem V.4.6 in such

a case, if g ∈ D

m+1,q

(Ω) and provided the term |g|

m+1,q

+ kgk

q,Ω

is added on

the right-hand side of the estimates (V.4.30)–(V.4.32). Notice that, unlike the case

where Ω is bounded, no compatibility condition is required between g and v

∗

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized Solutions 337

V.5 Existence, Uni queness, and L

-Esti mates:

q-generalized Solutions

In the present section we shall investigate the existence and uniqueness of

q-generalized solutions to system (V.4.1) and the validity of corresponding

estimates. As in Section V.3, we shall see that these results heavily depend

on how q and n are related. However, this time, if 1 < q ≤ n/(n −1) (1 < q <

n/(n − 1) for n = 2 ) the above mentioned solutions exist if and only if the

data satisfy a suitable compatibility condition; see (V.5.4). As a by-product,

our theory will furnish a general representation formula for functionals on

1,q

(Ω).

In order to simplify matters, we assume that the velocity ﬁeld v

∗

the boundary is identically zero. Generalizations to the more general non-

homog eneous case are left to the reader in Exercise V.5.1. We therefore con-

sider the following system

∆v = ∇p + f

∇ · v = 0

)

in Ω

v = 0 at ∂Ω,

(V.5.1)

We have

Deﬁnition V.5.1. A vector ﬁeld v : Ω → R

is called a q-generali z ed solution

to the Stokes system (V.5.1) if and only if v ∈ D

1,q

(Ω) and, furthermore,

(∇v, ∇ϕ) = −[f, ϕ], for all ϕ ∈ D

1,q

(Ω). (V.5.2)

Remark V.5.1 Unlike the deﬁnition of q-generalized solutions given in Sec-

tion V.1 for the Stokes problem (V.0.1), (V.0.2), in the case under considera-

tion q-generalized sol utions need not tend to zero a t inﬁnity; actually, as we

shall see, this happens if and only if n/(n−1) < q < n, see also Remark V.1.1.



From Lemma V.1.1 it follows tha t, provided Ω is l ocally Lipschitz and

f ∈ D

−1,q

(Ω), to any q-g eneralized v to (V.5.1) we can uniquely associate a

pressure ﬁeld p ∈ L

(Ω) such that

(∇v, ∇ψ) − (p, ∇ ·ψ) = −[ f , ψ], for all ψ ∈ D(Ω). (V.5.3)

As in the case of strong solutions, a fundamental role in our treatment is

played by exceptional q-generalized solutions, i.e., vector ﬁelds v ∈ D

1,q

(Ω)

solving (V.5.1) with f ≡ 0 (denoted from now on by (V.5.1)

). Their geometric

structure is characterized in the foll owing lemma.

338 V Steady Stokes Flow in Exterior Domains

Lemma V.5.1 Let Ω ⊂ R

be an exterior domain of class C

. Denote by S

the subspace of D

1,q

(Ω) ×L

(Ω) constituted by q-generalized solutions (v, p)

to (V.5.1)

. Then, if 1 < q < n (1 < q ≤ n for n = 2) S

= {0}, while if q ≥ n

(q > n for n = 2) dim(S

) = n.

Proof. Assume 1 < q < n. From Lemma II.6.2 in the limit |x| → ∞, it follows

that

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

Therefore, v is a q-generalized solutio n to the Stokes problem (V.0.1), (V.0. 2),

according to Deﬁnition V.1.1 corresponding to identically vanishing da ta and

so, in view of Theorem V.3.4, we have v ≡ 0 if 1 < q < n. Also, if q = n = 2,

from (V.5.2)

(∇v, ∇ϕ) = 0 for all ϕ ∈ D

1,2

(Ω)

and so we may take ϕ = v to obtain again v ≡ 0, which completes the proof

of the ﬁrst part of the lemma. Assuming next q ≥ n (q > n if n = 2), consider

the pairs (h

, π

) of solutions to (V.5.1)

constructed i n the proof of Lemma

V.4. 4. By what we have seen there, these solutions are linearly independent

and, moreover,

∈ D

1,q

(Ω) for all q > n/(n − 1).

Therefore, from Theorem II.7.6 and Theorem III.5.1,

∈ D

1,q

(Ω) for all q ≥ n (q > n if n = 2)

and the proof of the lemma is achieved. ut

Remark V.5.2 A basis {h

, π

} in S

can b e sometime explicitly exhibited.

For example, if Ω is exterior to a sphere, it is imm ediately seen that h

, π

can b e ta ken just as follows:

= e

−v

(i)

, π

= p

(i)

where v

(i)

, p

(i)

is the Stokes solutions (V.0.4), correspo nding to v

= e

, i =

1, 2, 3, respectively. Likewise if Ω is exterior to a circle, a basis is constituted

by the two independent solutions (V.0.7). 

Lemma V.5.1 has an important consequence, that is, a q-generalized solu-

tion to (V.5.1) with 1 < q ≤ n/(n − 1) (1 < q < n if n = 2) exist only if the

body force −f satisﬁes the compatibility condition

[f , h] = 0, for all (h, π) ∈ S

. (V.5.4)

In fact, condition (V.5.4) is al so suﬃcient to prove existence of q-generalized

solutions for the values of q speciﬁed above. In order to show this, we premise

the following general result that will be useful also for other purposes.

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized Solutions 339

Lemma V.5.2 Let u

, i = 1, . . ., N, be N independent functions in D

1,q

(Ω),

1 < q

< ∞. Then, the following properties hold.

(i) There exist N elements, `

, . . . , `

, of D

−1,q

(Ω), q = q

/(q

− 1), satis-

fying the conditions

, u

] = δ

, i, j = 1, . . . , N ,

and such tha t every f ∈ D

−1,q

(Ω) can be represented as follows

f = w +

i=1

[f , u

] `

where

[w, u

] = 0 , i = 1, . . ., N .

(ii) For any given f ∈ D

−1,q

(Ω) such that

[f, u

] = 0 , i = 1, . . ., N (V.5.5)

there exists a sequence, {f

} ⊂ C

∞

(Ω), whose elements satisfy

, u

) = 0 , for all m ∈ N and all i = 1, . . . , N , (V.5.6)

and, in addition,

lim

m→∞

|f − f

−1,q

= 0 .

Proof. We begin with a suitable decomposition of the space L

(Ω). Let g

i = 1, . . . , N, be independent elements of L

(Ω). We want to show that there

exist L

∈ C

∞

(Ω), i = 1, . . ., N, such that

, g

) = δ

, i, j = 1, . . . , N . (V.5.7)

The proof of (V.5 .7) will be given by induction.

Suppose N = 1. Then,

there exists ψ ∈ C

∞

(Ω) such that (ψ, g) 6= 0 , because, otherwise, g ≡ 0,

which contradicts the assumption. We then choose L = ψ/(ψ, g), thus proving

(V.5.7) for N = 1. Next, assume that there exist

∈ C

∞

(Ω), i = 1, . . ., N−1,

N ≥ 2, such that

(

, g

) = δ

, i, j = 1, . . . , N − 1 , (V.5.8)

then we show that there are L

∈ C

∞

(Ω), i = 1, . . . , N satisfying (V.5.7). In

fact, set

γ = −

N−1

j=1

(

, g

+ g

. (V.5.9)

I owe the proof of this property to Professor Christian Simader and Dr.

Thorsten Riedl.

340 V Steady Stokes Flow in Exterior Domains

Then, by the same token, there is ϕ ∈ C

∞

(Ω) such that

(ϕ, γ) 6= 0 , (V.5.10)

and so, setting

l = ϕ −

N−1

j=1

(ϕ, g

)

, (V.5.11)

and using the induction hypothesis (V.5.8), we ﬁnd

(l, g

) = (ϕ, g

) − (ϕ, g

) = 0 , k = 1, . . . , N − 1 , (V.5.12)

whereas, using (V.5.9) and (V.5 .10), we obtain

(l, g

) = (ϕ, g

) −

N−1

j=1

(ϕ, g

)(L

, g

)

= (ϕ, γ) +

N−1

j=1

, g

)(ϕ, g

) −

N−1

j=1

(ϕ, g

)(L

, g

)

= (ϕ, γ) 6= 0 .

(V.5.13)

Therefore, from (V.5.11)–(V.5.13), we deduce that, deﬁning

(l, g

)

, L

−(

, g

, i = 1, . . . , N −1 ,

the functions L

, . . ., L

obey property (V.5.7 ) that, consequently, is com-

pletely proved. Next, let F ∈ L

(Ω) be arbitrary, and set

W = F −

i=1

(F , g

where L

∈ C

∞

(Ω) satisfy (V.5.7). Then, clearly, in view of (V.5.7), we have,

on the one hand,

(W , g

) = 0 , i = 1, . . . , N , (V.5.14)

and, on the other hand,

F = W +

i=1

(F , g

, (V.5.15)

which, since W is uniquely determined, furnishes the desired decompositio n of

the space L

(Ω). Now, pick f ∈ D

−1,q

(Ω). From Theorem II.1.6 and Theorem

II.8.2 we know that there exists F ∈ L

(Ω) such that

[f , ψ] = (F , ∇ψ) . (V.5.16)

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized Solutions 341

Thus, if we choose, in particular, g

= ∇u

, the statements in part (i) of

the lemma follow from this representation and from (V.5.7), (V.5 .14), and

(V.5.15). We shall now show part (ii). Let f ∈ D

−1,q

(Ω), and let F be a

corresponding function in L

(Ω) satisfying (V.5.16). Since f must satisfy

(V.5.5), we ﬁnd that F obeys the following conditions

(F , ∇u

) = 0 , ı = 1, . . ., N . (V.5.17)

Denote by {F

} ⊂ C

∞

(Ω) a sequence such that F

→ F in L

, and set

= F

−

i=1

, ∇u

With the help of (V.5.7) and of (V.5.17), we easily establish the following

properties

} ⊂ C

∞

(Ω)

, ∇u

) = 0 , for all m ∈ N

and all k ∈ {1, . . ., N},

→ F in L

(Ω) .

The last statement of the lemma then follows by setting f

= ∇ · W

m ∈ N. ut

We are now in a position to prove the following.

Theorem V.5.1 Let Ω be an exterior do main in R

of class C

. Then, for

every f ∈ D

−1,q

(Ω) satisfying (V.5.4) if 1 < q ≤ n/(n−1) (1 < q < n/(n −1)

if n = 2) there exists one and only one q-generalized solution to (V.5.1) such

that

(v, p) ∈ D

1,q

(Ω) × L

(Ω) / S

Moreover, this solution veriﬁes

inf

(h,π)∈S

{|v − h|

1,q

+ kp − πk

} ≤ c|f|

−1,q

. (V.5.18)

Proof. As in the proof o f Theorem V.4.6, it is enough to show the result for

functions f ∈ C

∞

(Ω), that, when 1 < q ≤ n/(n − 1) (1 < q < n/(n − 1) if

n = 2), satisfy, in addition, the condition

(f, h

) = 0 , for all i = 1, . . . , n . (V.5.19)

In fact, the general case will b e a consequence of inequality (V.5.18), of the

density of C

∞

(Ω) into D

−1,q

(Ω) and of Lemma V.5.2. Thus, for a smooth f

we construct a solution v ∈ D

1,2

(Ω), p ∈ L

(Ω) by the methods employed in

Theorem V.2.1 (see also Remark V.2.1). This solution satisﬁes

342 V Steady Stokes Flow in Exterior Domains

|v|

1,2

+ kpk

≤ c |f|

−1,2

which, in particular, proves the theorem in the special case where q = n/(n −

1), n = 2. From Theorem IV.4.2 and Theorem IV.6.1 it follows that

v ∈ C

∞

(Ω) ∩ W

2,q

(Ω

), p ∈ C

∞

(Ω) ∩ W

1,q

(Ω

) (V.5.20)

for a ll R > δ(Ω

) and q > 1. Moreover, from Theorem V.3.2, we obtain

∇v ∈ L

(Ω

R/2

), p ∈ L

(Ω

R/2

)

for all q > n/(n − 1). This property, together with (V.5.20), with the aid of

Theorem II.7.1 and Theorem III.5.1 in turn implies

v ∈ D

1,q

(Ω), p ∈ L

(Ω), q > n/(n − 1). (V.5.21)

Applying Theorem IV.2.2 to system (V.4.3)–(V.4.4) we readily deduce

k∇vk

q,Ω

R/2

+ kpk

q,Ω

R/2

≤ c (|ϕf + T (v, p) · ∇ϕ|

−1,q

+ kv|∇ϕ|k

) . (V.5 .22)

Since q

< n, from Sobolev i nequality (II.3.7) and for all Φ ∈ D

1,q

)

|(ϕf, Φ)| ≤ |f|

−1,q

|ϕΦ|

1,q

≤ c|f|

−1,q

|Φ|

1,q

. (V.5.23)

Likewise, taking into account that Φ

ϕ ∈ W

1,q

(Ω

), i, j = 1, . . . , n, from

(V.3.1) we have

|(T (v, p) · ∇ϕ, Φ)| ≤

i,j=1

(| (D

, Φ

ϕ)| + |(D

, Φ

ϕ)|

+ |(pδ

, Φ

ϕ)|)

≤ c



kvk

q,Ω

R/2

+ kpk

−1,q,Ω

R/2



|Φ|

1,q

(V.5.24)

Therefore, (V.5 .22)–(V.5.24) imply

k∇vk

q,Ω

R/2

+ kpk

q,Ω

R/2

≤ c



|f|

−1,q

+ kvk

q,Ω

R/2

+ kpk

−1,q,Ω

R/2



. (V.5.25)

Moreover, from estimate (IV.6.8) in Ω

we also obtain

kvk

1,q,Ω

+ kpk

q,Ω

≤ c



|f|

−1,q

+ kvk

q,Ω

+ kpk

−1,q,Ω

+ kvk

1−1/q,q(∂B

)



(V.5.26)

where we used the obvious inequality

kf k

−1,q,Ω

≤ |f |

−1,q

By employing the traceTheorem II.4.4 at the boundary term i n (V.5.26) we

deduce

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized Solutions 343

kvk

1,q,Ω

+ kpk

q,Ω

≤ c



|f|

−1,q

+ kvk

q,Ω

+ kpk

−1,q,Ω

+ k∇vk

q,Ω

R/2



(V.5.27)

The solution v, p will then satisfy (V.5.21) and, by (V.5.25), (V.5.27), the

inequality

|v|

1,q

+ kpk

≤ c ( |f |

−1,q

+ kvk

q,Ω

+ kpk

−1,q,Ω

) (V.5. 28)

for all q > n/(n − 1). Let us show that if f satisﬁes (V.5.19) the properties

just shown continue to hold when 1 < q ≤ n/(n − 1) (1 < q < n/(n − 1) if

n = 2). We already know that v and p satisfy (V.5.20) for all q > 1. Also,

v and p obey the asymptotic expansion (V.3.17), (V. 3.18), and (V.3.19). If

n > 2, since v ∈ D

1,2

(Ω) we ﬁnd v

∞

= 0, and so to show v ∈ D

1,q

(Ω),

1 < q ≤ n/(n − 1), by Theorem II.7.1 it is necessary and suﬃcient to prove

that the vector T deﬁned in (V.3 .20) is zero. Likewise, for n = 2, since it is

readily shown that v ∈ D

1,2

(Ω) implies T = 0 (see Remark V.3.5), to prove

v ∈ D

1,q

(Ω), 1 < q < n/(n − 1), again by Theorem II.7.1 it is necessary and

suﬃcient to prove v

∞

= 0. From Green’s formula applied in Ω

we have for

all R > δ(Ω

)

−

Ω

f · h

∂B

· T (v, p) · n − v · T (h

, π

) · n}. (V.5.29)

By this relation and the asymptotic properties of (h

, π

) (see Lemma V.4 .4),

and of (v, p) (see Theorem V.3.2) it follows for n > 2

Ω

f · h

= −e

∂B

T (v, p) ·n

∂B

{(e

−h

) · T (v, p) ·n + v · T (h

, π

) · n}

= −e

∂B

T (v, p) ·n + O(1/R

n−2

)

and so, by (V.5.19),

∂B

T (v, p) ·n = O(1/R

n−2

), i = 1, 2, 3. (V.5. 30)

On the other hand, by taki ng Ω

so that Ω

∩supp (f ) = ∅, from (V.5.1) we

have

T ·e

= e

∂B

T (v, p) · n = O(1/R

n−2

which entails T = 0, thus proving v ∈ D

1,q

(Ω), p ∈ L

(Ω), 1 < q ≤ n/(n−1),

for n ≥ 3. Suppose now n = 2. As already noticed T = 0 for solutions

v ∈ D

1,2

(Ω), p ∈ L

(Ω) and so from (V.3.17)– (V.3.19) it comes out that

T ( v, p) = O(1/|x|

)

v = v

∞

+ O(1/|x|).

344 V Steady Stokes Flow in Exterior Domains

Also, by (V.4.27) we have

= O(log |x|)

T (h

, π

) = O(1/| x|).

Consequently, (V.5.29) furnishes for all ε ∈ (0, 1)

∞

∂B

T ( h

, π

) · n = O(1/R

1−ε

). (V.5.31 )

However, a comparison between the general expansion formulas (V.3.17)–

(V.3.21) and (V.4. 27) reveals

∂B

T (h

, π

) · n = e

which once replaced into (V.5.31) yields v

∞

= 0. Therefore, we conclude

v ∈ D

1,q

(Ω), p ∈ L

(Ω), 1 < q ≤ n/(n − 1) if n = 2. We shall next establish

the validity of (V.5.28 ). As in the case where q > n/(n − 1) we arrive at

inequality (V.5.22). Now, ta king into account that ϕ = 0 near ∂Ω and ϕ = 1

in Ω

we have

Ω

(ϕf + T (v, p) · ∇ϕ)

Ω

ϕ(f − ∇ · T ) +

∂Ω

ϕT · n +

∂B

ϕT ·n =

∂B

T · n

and so, by what we have shown,

Ω

= 0. (V.5.32)

By (V.5.32 ), in view of Theorem II.8.1 (see also Remark II.8.1), the f unctional

[Φ] ∈ D

1,q

) → (f

, Φ) , Φ ∈ [Φ] , q

≥ n

is well deﬁned and independent of the particular choice of the function Φ in

the equivalence class [ Φ] . Thus, for any such Φ, we deﬁne

Φ = Φ −

|Ω

Ω

Φ,

so that by Poincar´e’s inequality (II.5.10)

Φk

,Ω

≤ c|Φ|

1,q

. (V.5.33)

From (V.5.32) it follows that

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized Solutions 345

, Φ) = (f

Φ) = (ϕf,

Φ) + (T (v, p) · ∇ ϕ,

Φ),

and so we may proceed as in (V.5. 23), (V.5.24) by using this time (V.5.33)

instead of the Sobolev inequality to reach estimate (V.5.25). Since (V.5.26)

holds for all q > 1, we may ﬁnally establish, in the same way as in the case

where q > n/(n −1), the validity of (V.5.28). Once (V.5.28) is recovered, we

obtain from it

inf

(h,π)∈S

{|v −h|

1,q

+ kp − πk

}

≤ c



|f|

−1,q

+ inf

(h,π)∈S

[|v − h|

q,Ω

+ kp − πk

−1,q,Ω

]



(V.5.34)

Using a contradiction argument entirely ana logous to that of Theorem V.4. 6

and based on compactness results of Exercise II.5.8 and Theorem II.5.3, we

can show

inf

(h,π)∈S

{|v −h|

q,Ω

+ kp − πk

−1,q,Ω

} ≤ c

|f |

−1,q

which, o nce replaced into (V.5 .34), yi el ds (V.5.18). Existence is then fully

carried out. The uniqueness of solutions just determined is also immediately

established and therefore the proof of the theorem is complete. ut

A signiﬁcant consequence of Theorem V.5.1 is a general representation

of functionals on the space D

1,q

(Ω), 1 < q < ∞. Speciﬁcally, we have the

following result; see Galdi & Simader (1990, Section 7).

Theorem V.5.2 Let Ω be as in Theorem V.5.1, and let f ∈ D

−1,q

(Ω). The

following properties hold.

(i) If q > n/(n − 1) (q ≥ n/(n − 1) if n = 2) then f can be represented as

[f , ϕ] = (∇v, ∇ϕ), ϕ ∈ D

1,q

(Ω), (V.5.35)

with v uniquely determined if q < n (q ≤ n if n = 2), while v is deter-

mined up to a function h, if q ≥ n (q > n if n = 2), where (h, π) ∈ S

(ii) If 1 < q ≤ n/(n −1) (1 < q < n/(n −1), if n = 2), there exist a uniquely

determined vector function v ∈ D

1,q

(Ω), and n functions r

, . . ., r

with

∈ D

1,q

(Ω), i = 1, . . ., n, uniquely determined up to a constant, such

that for all ϕ ∈ D

1,q

(Ω) and for any ﬁxed basis {h

, π

} in S

we have

[f , ϕ] = (∇v, ∇ϕ) +

i=1

[f, h

](∇r

, ∇ϕ).

Proof. Since D

1,q

(Ω) is a subspace of D

1,q

(Ω), by the Hahn- Banach Theorem

II.1.7(a), there exists F ∈ D

−1,q

(Ω) such that [F , ϕ] = [f, ϕ], for a ll ϕ ∈