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

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246 IV Steady Stokes Flow in Bounded Domains

for all i, j = 1, . . . , n. From Theorem II.7. 6 and the Calder´on–Zygmund the-

orem it is easy to show that, for any i and j, the function φ

≡ D

∗ ϕ

belongs to D

1,q

) and that

k∇φ

≤ c

kϕk

where c

= c

(n, q). From this inequality, (IV.2.28), and the fact that f

satisﬁes (II.8.10) if q

≥ n, we deduce

k∇v

q,B

≤ c

|f |

−1,q

which, si nce c

is independent of ρ, in the limit ρ → ∞ yields

1,q

≤ c

|f |

−1,q

. (IV.2.29)

with c

= c

(n, q). As a consequence, (IV.2.25 ) follows f rom (IV.2.27),

(IV.2.29), and Lemma IV.2.1, and the existence proo f is accomplished. It

is worth empha sizing that the soluti on v, p just considered for f, g ∈ C

∞

)

is a smooth solutio n to (IV.2.7) and that it satisﬁes

v ∈ D

1,r

), p ∈ L

) for all r ∈ (1, ∞).

With this in mind, we shall now show the uniqueness part. Let v

be a q

generalized solution to (IV.2.7), corresponding to the same f and g. Setting

w ≡ v

−v,

from the deﬁnition of s-generalized solution it follows that

(∇w, ∇φ) = 0, for all φ ∈ D

1,q

) ∩ D

1,q

)

(w, ∇ϕ) = 0, for all ϕ ∈ C

∞

(IV.2.30)

By what we have shown, given F ∈ C

∞

), corresponding to f ≡ F , g ≡

φ ≡ 0 there exists a smooth solution u, τ to (IV.2.7 ), which further satisﬁes

(u, τ) ∈ D

1,r

) × L

), for all r ∈ (1, ∞). (IV.2.31)

If r ≤ n/(n − 1), the function F must verify (II.8 .10). We now multiply

(IV.2.7)

, written for u and τ , by ψ

w where ψ

is the Sobolev “cut-oﬀ”

function deﬁned in (II.7.1). Integrating by parts over R

, with the help of

Exercise II.4.3 , we deduce for all suﬃciently large R:

∇u : ∇w = −

(∇ψ

·∇u · w − τ ∇ψ

w) − (F , w). (IV.2.32)

By the H¨older inequality, we have

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 247



(∇ψ

·∇u · w − τ ∇ψ



≤



|u|

1,q

+ kτ k



k∇ψ

Ω

+ (|u|

1,q

+ kτk

) k∇ψ

Ω

(IV.2.33)

where

Ω

is deﬁned in (II.7.3) a nd contains the support of ∇ψ

. As shown

in the proof of Theorem I I .7.1,

k∇ψ

Ω

+ k∇ψ

Ω

→ 0 as R → ∞ (IV.2.34)

and so, letting R → ∞ into (IV.2.32), from (IV.2.31), (IV.2.33), and (IV. 2.34)

it follows that

(∇u, ∇w) = (F , w). (IV.2.35)

Because of (IV.2.30), we may now take ϕ = u into (IV.2.30) and use (IV.2.35)

to ﬁnd

(F , w) = 0, (IV.2.36)

which, by the a rbitrarity of F , i n turn implies w ≡ 0 a.e. in Ω, if both q

and q are strictly less than n. Otherwise, since F has to satisfy (II.8.10), we

obtain w =const. a.e. in Ω. From (IV.2.24) we then recover τ = const. a.e.

in Ω, which compl etes the proof of the theorem. ut

IV.3 Existence, Uniqueness, and L

-Esti mates in a

Half-Space. Evaluation of Green’s Te nsor

In this section we shall prove results simil ar to those of Theorem IV.2.1 and

Theorem IV.2.2 for the inhomogeneous Stokes problem in the half-space R

n ≥ 2. Here the situation is complicated by the fact that the domai n has a

boundary, even if a simple one. We begin to study the probl em

∆W = ∇S

∇ · W = 0

)

in R

W = Φ at Σ ≡ {x ∈ R

: x

= 0},

(IV.3.1)

where Φ ∈ C

(Σ) for some m ≥ 1, Φ = O(log |ξ|) as |ξ| → ∞,

and D

Φ ∈

C(Σ), 1 ≤ |α| ≤ m. To this end, we introduce with Odqvist (193 0, §2) the

Stokes double-layer potentials (for the half-space)

(x) = 2

(y)



−δ

(x − y) +

∂U

(x − y)

∂y

∂U

(x − y)

∂y



dσ

S(x) = −4

(y)

∂q

(x − y)

∂y

dσ

(IV.3.2)

We could allow Φ to “grow” faster. Such a weaker assumption, however, would

be unessential for further purposes.

248 IV Steady Stokes Flow in Bounded Domains

where n (= −e

) is the outer normal to Σ.

Recalling the expressions (IV.2.3)

and (IV.2 .4) of the Stokes fundamental solution, (IV.3.2) can be rewritten as

(x) =

−y

, x

)Φ

)dy

S(x) = −D

k(x

−y

, x

)Φ

)dy

(IV.3.3)

with z

= (z

, . . . , z

n−1

) and

− y

, x

) =

−y

)(x

− y

)

(|x

−y

+ x

)

(n+2)/2

, y

= 0,

k(x

− y

, x

) =

nω

(|x

−y

+ x

)

n/2

, y

= 0.

(IV.3.4)

We easily show that W and S are C

∞

solutions to (IV.3.1 )

1,2

. In fact, it is

clear that W and S are smooth; in addition, since q is harmonic (for x 6= y)

from (IV.2.5) and (IV.3.2)

we ﬁnd

∆W

(x) = −2

(y)



∂

(x − y)

∂y

∂x

∂

(x − y)

∂x

∂y



dσ

and, by (IV.2 .2)

∂W

(x)

∂x

= 2

(y)



−δ

∂q

(x − y)

∂x

∂

(x − y)

∂x

∂y

∂

(x − y)

∂x

∂y



dσ

= −2

(y)

∂q

(x − y)

∂x

dσ

However, it is immediately checked that

∂q

∂x

∂q

∂x

∂q

∂y

∂q

∂y

∂q

∂y

= −

∂q

∂x

∂q

∂x

∂q

∂y

= 0 , x 6= y ,

and so,we deduce that (IV.3.1)

1,2

are satisﬁed. Also, for all x

∈ R

n−1

we can

prove

lim

→0

W (x

, x

) = Φ(x

). (IV.3.5 )

Actually, fo r ﬁxed ξ ∈ R

n−1

, we take an n − 1-dimensional bal l C

, centered

at ξ such that

sup

y∈C

|Φ(ξ) − Φ(y)| < ε. (IV.3.6)

By a direct calculation based on (IV.3.4)

, one shows that the fol lowing rela-

tions hold:

The functions (IV .3.2) are the analogue of the familiar Poisson integral for the

Dirichlet problem for Laplace equation in the half-space, c onsidered at the end

of Section II.11.

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 249

(i)

(ξ − y

, x

)dy

= δ

+ o(1) as x

→ 0,

(ii)

(ξ − y

, x

)|dy

≤ c

with c independent of x

and ξ. Likewise, using the growth properties of Φ,

we obtain

(iii)

Σ−C

(ξ − y

, x

)Φ

)dy

= o(1) as x

→ 0.

Therefore, using (i) and (iii) we recover as x

→ 0

(ξ, x

) − Φ

(ξ) =

(ξ − y

, x

)Φ

)dy

− Φ

(ξ)

Σ−C

(ξ − y

, x

)Φ

)dy

(ξ − y

, x

)[Φ

) − Φ

(ξ)] dy

+ o(1),

which, in view of (IV.3. 6) and (ii), in turn implies

lim sup

→0

|W (ξ, x

) −Φ(ξ)| ≤ c ε.

By the arbitrarity of ε, we deduce (IV.3.5).

We wish to determine some L

-estimates for W and S in terms of Φ

analogous to those derived for the Dirichlet problem for the Laplace equation

at the end of Section II.11, and which for problem (IV.3.1) were proved for

the ﬁrst time by Cattabriga (1961). To be speciﬁc, we shall deal with the case

n = 3, the general case being treated sim ilarly. For | α| ≤ m, from (IV.3 .3) we

have

0α

(x) =

− y

, x

0α

)dy

0α

S(x) = −D

k(x

− y

, x

0α

)dy

(IV.3.7)

where

0α

≡

∂

|α|

∂x

. . .∂x

n−1

, α

+ α

+ . . . + α

n−1

= |α|. (IV.3.8)

We o bserve that from (IV .3.4) it foll ows that K

, x

), i, j = 1, 2, 3, and

k(x

, x

) are of class C

∞

for x

> 0, with bo unded derivatives of any order in

the hemisphere {|x|

= 1, x

> 0}. Moreover, since

250 IV Steady Stokes Flow in Bounded Domains

, x

) =

2π

|x|

≡

Ω



|x|



|x|

k(x

, x

) =

|x|

≡



|x|



|x|

and observing that

Ω

, 0) = ω(x

, 0) = 0 for all x

6= 0 ,

we conclude that K

and k satisfy all assumptions of Theorem II.11.6. So, if

Φ is in L

(Σ) and has ﬁnite hhD

Φii

1−1/q,q

norm, 1 < q < ∞, we obtain

∇D

0α

W , D

0α

S ∈ L

)

together with the following estimate

0α

W |

1,q

+ kD

0α

≤ c hhD

Φii

1−1/q,q

, (IV.3.9)

where c depends only on q, n (=3), and α. A similar i nequality also valid for

-derivatives can be easily obtained using (IV.3.9) and the fact that W , S

is a solution to (IV.3.1 ). Let us consider the case |α| = 1. D iﬀerentiati ng

(IV.3.1)

with respect to x

and employing (IV.3.9), we have

≤ kD

+ kD

≤ 2chh∇Φii

1−1/q,q

. (IV.3.10)

Moreover, setting

= (W

, W

), ∆

∂

∂x

∂

∂x

, ∇



∂

∂x

∂

∂x



from (IV.3.1)

we obtain

∆W

= D

= −∆

+ ∇

and thus, agai n by (IV.3.9),

+ kD

≤ 8chh∇Φii

1−1/q,q

which, along with (IV.3.10), proves the desired estima te; that is,

|W |

2,q

+ |S|

1,q

≤ chh∇Φii

1−1/q,q

where c depends only on n, α, and q. More generally, if |α| ≥ 1, one can use

a similar argum ent (which we leave to the reader) to show the inequality

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 251

|W |

k+1,q

+ |S|

k,q

≤ c

|α|=k

hhD

Φii

1−1/q,q

, (IV.3.11)

where 0 ≤ k ≤ m. I f q ∈ (1, n) we may obtain a sharper estimate. In fact,

from Theorem II.6.3(i) and (IV.3.11), we deduce, in particular, the existence

of constant vectors W

(α)

such that

|α|=k

W − W

(α)

nq/(n−q)

≤ c

|α|=k

hhD

Φii

1−1/q,q

However, by direct inspection, from (IV.3.3 )

and (IV.3.5)

, we have D

W →

0 as x

→ ∞, |α| ≥ 0 which, in turn, implies (as the reader will easily show)

(α)

= 0. Therefore, we conclude

|W |

k,nq/(n−q)

+ |W |

k+1,q

+ |S|

k,q

≤ c

|α|=k

hhD

Φii

1−1/q,q

, if q ∈ (1, n) .

(IV.3.12)

The results proved so far continue to hold if we weaken somewhat the

regularity assumptions m ade on Φ. For later purposes (see Section IV.5) it is

interesting to consider the case when

Φ ∈ W

m,q

(Σ) with

|α|=m

hhD

Φii

1−1/q,q

< ∞.

Under these hypotheses it is simple to show that (IV.3.2) is still a C

∞

-solution

of (IV.3.1)

1,2

though, of course, the boundary value is now attained a priori

in a way less regular than (IV.3.5), that is (see Exercise IV.3.1),

lim

→0

|W (x

, x

) −Φ(x

= 0, (IV.3. 13)

where C is any compact subset of Σ. Furthermore, since by the results of

Exercise II.10. 1 it follows that hhD

Φii

1−1/q,q

is ﬁnite for all |α| ∈ [0, m], we

conclude that

∇W , D

S ∈ L

and that inequality (IV.3.11) holds for these values of α.

We may thus summarize all previous results i n the following.

Lemma IV.3.1 Let Φ ∈ C

(Σ), m ≥ 1, with Φ(ξ) = O(log |ξ|) as |ξ| →

∞ a nd D

Φ ∈ C(Σ), 1 ≤ |α| ≤ m. Then the functions W , S deﬁned by

(IV.3.3), (IV.3.4) are of class C

∞

in R

and satisfy there (IV.3.1) and (IV.3.5 ).

Moreover, if Φ ∈ D

k,q

(Σ) and

|α|=k

hhD

Φii

1−1/q,q

is ﬁnite for some integer

k ∈ [0, m], 1 < q < ∞, then

(i) |W |

k+1,q

and |S|

k,q

are ﬁnite, and, if q ∈ (1, n), also |W |

k,nq/(n−q)

ﬁnite ;

252 IV Steady Stokes Flow in Bounded Domains

(ii) W , S satisfy i nequalities (IV.3.11), (IV.3.12).

Likewise, let

Φ ∈ W

m,q

(Σ) with

|α|=m

hhD

Φii

1−1/q,q

< ∞.

Then W , S sati sfy (IV.3.1), (IV.3.13) and statements (i) and (ii) hold for all

integers k ∈ [0, m].

Exercise IV.3.1 Show the validity of condition (IV.3.13). Hint: Use the same ar-

guments adopted in the proof of (IV.3.5).

We shall next consider the problem

∆w = ∇s + f

∇· w = g

)

in R

v = 0 at Σ,

(IV.3.14)

where f, g ∈ C

∞

) and shall prove the existence of a solution verifying

suitable estimates in terms of the data. This will be done by reducing (IV.3.14)

to (IV.3.1) and then using Lemma IV.3.1 . First, we make extensions f

and

of f and g to the whole of R

in the way suggested in Exercise II.3.10, so

that f

, g

∈ C

r+1

) for suﬃciently large r and

q,R

≤ ckD

q,R

≤ ckD

q,R

0 ≤ |β| ≤ r + 1, (IV.3.15)

where c depends only on r, n, and q. Successively, we look for a solution of

the form

w = w

W , s = s

S , (IV.3.16)

where w

, s

is the solution to (IV.2.7) with f ≡ f

and g ≡ g

, and whose

existence is ensured by Theorem IV.2.1, while

W ,

S solve

∆

W = ∇

∇ ·

W = 0







in R

W = −w

at Σ.

(IV.3.17)

We shall show tha t Φ ≡ −w

satisﬁes the assumptions of the ﬁrst part

of Lemma IV.3.1. Therefore, by that lemma,

W ,

S will obey, in particular,

estimate (IV.3.11). From Theorem IV.2.1 it follows that Φ ∈ C

∞

(Σ). More-

over, Φ(ξ) = O(log |ξ|) and D

Φ ∈ C

(Σ) for all |α| ∈ [1, s] if n = 2, while

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 253

Φ ∈ C

(Σ) for all m ∈ [0, s] if n = 3. Let us now prove that for any |α| ∈ [1, s]

and any q > 1

Φ ∈ L

(Σ),

hhD

Φii

1−1/q

< ∞,

(IV.3.18)

so that a ll the hypotheses of the ﬁrst part o f Lemma IV.3.1 are fulﬁlled.

Recalling that w

is given by (IV.2.8)

with F ≡ f

−∆h

= f

−∇g

where

is given by (IV.2.10) with g

in place of g, from (IV.2.12)

we ﬁnd

Φ(x

) = O(|x

−n+1

) as |x

| → ∞.

This property, along with the fact that D

Φ ∈ C

(Σ), implies (IV.3.18)

. On

the other hand, using Theorem II.10.2 and Theorem IV.2.1, it follows that,

for a ll |α| ≥ 0,

hhD

∇w

1−1/q,q

≤ c k D

∇w

q,R

≤ c (kD

q,R

+ kD

q,R

) .

(IV.3.19)

We may thus apply Lemma IV.3.1 and use (IV.3.16), (IV.3.19), (IV.3.15), and

Theorem IV.2.1 to obtain for all ` ≥ 0

|w|

`+2,q

+ |s|

`+1,q

≤ |w

`+2,q

+ |s

`+1,q

+ |

W |

`+2,q

+ |

`+1,q

≤ c (|f|

`,q

+ |g|

`+1,q

) ,

(IV.3.20)

where c depends only on q, `, and n. As in the proof of the previous lemma,

we observe that, if q ∈ (1, n), from Theorem II.6.3, the fact that D

w → 0 as

→ ∞, |α| ≥ 0, and (IV.3.20), we deduce that |w|

`+1,nq/(n−q)

is ﬁnite and

that

|w|

`+1,nq/(n−q)

≤ c (|f|

`,q

+ |g|

`+1,q

) . (IV.3.21)

By using exactly the same procedure employed for Theorem IV.2.1, based

on the density of C

∞

) in W

l,t

), inequalities (IV.3.20) and (IV.3.21),

and the functional properties of spaces

m,q

), we can extend the results

just proved to the case where

f ∈ W

m,q

), g ∈ D

m+1,q

) , m ≥ 0 , 1 < q < ∞. (IV.3.22)

Clearly, the corresponding solution, that we continue to denote by w, s, is

such that

w ∈

`=0

`+2,q

), s ∈

`=0

`+1,q

), (IV.3.23)

and, if q ∈ (1, ∞),

w ∈

`=0

`+1,nq/(n−q)

) . (IV.3.24)

254 IV Steady Stokes Flow in Bounded Domains

Thus, i n particular, by Lemma II.6.1,

w ∈ W

m+2,q

(C), s ∈ W

m+1,q

(C),

for all (open) cubes C ⊂ R

. Final ly, w and s satisfy (IV.3.20), and, if q ∈

(1, n), (IV.3.21) for all ` ∈ [0, m], and w has zero trace at the boundary.

We have therefore proved

Lemma IV.3.2 Fo r any f, g in the class deﬁned by (IV.3.22), there exists a

solution w, s a.e. to the nonhomogeneous Stokes system (IV.3.14) such that

w ∈ W

m+2,q

(C), s ∈ W

m+1,q

for all open cubes C ⊂ R

. Moreover, w, s satisfy (IV.3.23) and, if q ∈ (1 , n),

also (IV.3.24). In addi tion, for every ` ∈ [0, m], the f ollowing inequality holds:

|w|

`+2,q

+ |s|

`+1,q

≤ c (|f|

`,q

+ |g|

`+1,q

) . (IV.3.25)

If q ∈ (1, n), we also have:

|w|

`+1,nq/(n−q)

+ |w|

`+2,q

+ |s|

`+1,q

≤ c (|f|

`,q

+ |g|

`+1,q

) . (IV.3.26)

In the above inequalities, c = c(n, `, q).

The next step is to prove uniqueness for such solutions. In this respect,

the general result proved in the fol lowing theorem is appropriate.

Theorem IV.3.1 Let u ∈ W

m+2,q

(C), π ∈ W

m+1,q

open cube in R

, n ≥ 2) be a solution a.e. to the Stokes system (IV.3.14)

with f ≡ g ≡ 0. Assume |u|

`+2,q

ﬁnite fo r some ` ≥ 0 and some q ∈ (1, ∞).

Then

|u|

`+2,q

= |π|

`+1,q

= 0.

In particular, if ` = 0, then

u = a x

, π = const.

with a = (a

, . . . , a

n−1

, 0) constant vector.

Proof. To ﬁx the ideas, we shall consider the case n = 3 and ` = 0, the general

case being handled in a completely analogous way. As in the proof of Theorem

IV.2. 1, we obtain at once that π is harmonic througho ut the half-space. Let

and π

be the regularizatio ns of u a nd π, respectively, with respect to

= (x

, x

). It is readily shown that, fo r all ε

> 0, u

and π

satisfy the

same boundary-value problem as u, π and that

w ≡ D

, s ≡ D

with D

deﬁned in (IV.3 .8), are solutions to the following problem

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 255

∆w = ∇s

∇ · w = 0

)

in R

w = 0 at Σ ≡ R

× {0}.

(IV.3.27)

By assumption and the properties of molliﬁers one shows

w, s, D

w, ∇s ∈ L

)

and, therefore, by a simple interpolation,

w ∈ W

2,q

);

(see Exercise IV.3.2). Let now f ∈ C

∞

) and let v, p be the corresponding

solution determined i n Lemma IV.3.2 with g ≡ 0. Evidently,

≡ D

v, p

≡ D

solve (IV.3.14) with f replaced by D

f a nd g ≡ 0. Moreover,

∈ W

2,q

), p

∈ W

1,q

We next multiply (IV.3.27 )

by v

, integrate by parts, and use Theorem III.1.2

to obtain

0 =

w · D

f =

w · f.

Since f is arbitrary from C

∞

) and w ∈ L

), from this relation and

(IV.3.27) we derive, in parti cular, for all x ∈ R

and all ε

> 0

w(x) ≡ D

(x) = 0

(x) = c

(IV.3.28)

where c

depends only on ε

. Since π

is harmo nic throughout R

with ∇ π

∈

) we deduce with the help of (IV.3.28)

that π

must be constant

with respect to x, for all ε

> 0, which ﬁnally gives π = const. This being

established, from

∆u

= ∇π

it follows that u

is harmoni c throughout the half-space. Recalling that

u ∈ L

) and u

= 0 at Σ, from the prop erty of molliﬁers and R e-

mark II.11.3 it is easy to show that D

u ≡ 0, i.e., u = u

+ U

·x, where u

and U

are a constant vector and a constant 3×3 matrix, respectively. Since u

is zero at Σ and ∇·u = 0 in R

one obtains, in conclusion, u = x

, a

, 0),

for som e a

, a

∈ R. The proof of the theorem is thus complete. ut