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

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256 IV Steady Stokes Flow in Bounded Domains

On the strength of the results shown so far, we can prove the following

theorem, which represents the anal ogue of Theorem IV.2.1 for the half-space.

Theorem IV.3.2 Let Σ = {x ∈ R

: x

= 0}. For every

f ∈ W

m,q

), g ∈ D

m+1,q

)

and

Φ ∈ W

m+1,q

(Σ) with

|α|=m+1

hhD

Φii

1−1/q,q

ﬁnite,

m ≥ 0, 1 < q < ∞, n ≥ 2, there exists a pair of functions v, p such that

v ∈ W

m+2,q

(C), p ∈ W

m+1,q

(C),

for all op en cubes C ⊂ R

, solving a.e. the following nonhomogeneous Stokes

system

∆v = ∇p + f

∇ · v = g

)

in R

v = Φ at Σ.

(IV.3.29)

Moreover, for all ` ∈ [0, m], the seminorms |v|

`+2,q

and |p|

`+1,q

are ﬁnite and

we have

|v|

`+2,q

+ |p|

`+1,q

≤ c



|f|

`,q

+ |g|

`+1,q

|α|=`+1

hhD

φii

1−1/q,q



. (IV.3.30 )

If, in particular, q ∈ (1, n), then also |v|

`+1,nq/(n−q)

is ﬁnite, and, for all

` ∈ [0, m], we have

|v|

`+1,nq/(n−q)

+|v|

`+2,q

+|s|

`+1,q

≤ c



|f |

`,q

+|g|

`+1,q

|α|=`+1

hhD

φii

1−1/q,q



(IV.3.31)

In the above inequalities c = c(n, q, m). Furthermore, if v

, p

is another

solution to (IV.3.29) corresponding to the same data and, for some ` ∈ [0, m],

`+2,q

is ﬁnite, then |v − v

`+2,q

= |p − p

`+1,q

= 0. In particular, if

` = 0, there exists a vector a = (a

, . . . , a

n−1

, 0) such that v = v

+ ax

p = p

+const. Finally, if ` = 0, q ∈ (1, n) and |v

1,nq/(n−q)

is ﬁnite,

then

a = 0.

Proof. A solution to (IV.3.29) is gi ven by v = W + w, p = S + s, where

W , S and w, s are given in L emma IV.3.1 and Lemma IV.3.2. Then, from

those lemmas, we deduce that v, p possesses all the properties claimed in the

Notice that, by T heorem II.6.3(i), we can ﬁnd a uniquely determined c ∈ R

such

that k∇v

+ ck

nq/(n−q)

is ﬁnite. So, the request is that c = 0.

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 257

statement. This concludes the proof of existence. Concerning uniqueness, in

view of Theorem IV.3.1 we have only to discuss the case when a = 0. But this

is obvious, since, under the stated assumptions, ∇(v − v

) ∈ L

nq/(n−q)

The proo f of the theorem is complete. ut

Exercise IV.3.2 Let u ∈ L

) with D

u ∈ L

), 1 < q < ∞. Show that

∇u ∈ L

). Hint: Use Ehrling’s inequality (II.5.20) on every unitary cube in R

Our next task i s to prove existence of q-generalized solutions to problem

(IV.3.29). In complete analo gy with Deﬁnition IV.2.1, by this latter we mean

a ﬁeld v such that

(i) v ∈ D

1,q

);

(ii) (∇v, ∇ϕ) = −[f, ϕ], for all ϕ ∈ D

1,q

);

(iii) (v, ∇ϕ) = −(g, ϕ), for all ϕ ∈ C

∞

);

(iv) v obeys (IV.3.29)

in the trace sense (see (IV.3.13)).

In view of Lemma IV.1.1, i f f ∈ W

−1,q

(C), for all open cubes C ⊂ R

then, to every q-generalized solution we may associate a pressure ﬁeld p ∈

loc

) such that

(∇v, ∇ψ) − (p, ∇· ψ) = −[f , ψ], for all ψ ∈ C

∞

Moreover, if f ∈ D

−1,q

), then p ∈ L

We have

Theorem IV.3.3 Given

f ∈ D

−1,q

), g ∈ L

)

and

Φ ∈ L

(Σ) with hhΦii

1−1/q,q

ﬁnite,

1 < q < ∞, n ≥ 2, there exists at least o ne q-generalized solution to (IV.3.29).

Moreover, this solution satisﬁes the inequality

|v|

1,q

+ kpk

≤ c



|f |

−1,q

+ kgk

+ hhφii

1−1/q,q



, (IV.3.32)

where the constant c depends o nly on q and n. Finally, if v

is a q

-generalized

solution (1 < q

< ∞, q

possibly diﬀerent from q) corresponding to the same

f, g, and Φ, it follows that v

≡ v a.e. in R

and, consequently, denoting

by p

the pressure ﬁeld associated to v

by Lemma IV.1.1, we also have p

≡

p + const a.e. in R

Proof. We ﬁrst show existence. As we know, if f , g ∈ C

∞

), a smooth

solution to (IV.3.29 ) is given by

v = v

+ W , p = p

+ S,

258 IV Steady Stokes Flow in Bounded Domains

where

(x) = h

(x) + U

∗ (f

−D

)(x) ≡ h

(x) + A

(x)

(x) =

−y

, x

, 0)dy

− y

, x

)[Φ

) −h

, 0)]dy

≡ B

(x) + b

(x)

(x) = −q

∗ (f

− D

)(x)

S(x) = D

k(x

−y

, x

, 0)dy

−

k(x

− y

, x

)[Φ

) − h

, 0)]dy

Moreover, h is deﬁned in (IV.2.10) while f

and g

are sm ooth extensions of

f and g to R

satisfying (IV.3.15). Since h(y

, 0) ∈ L

(Σ), by Lemma IV.3.1

we ﬁnd

|b|

1,q

≤ c



hhφii

1−1/q,q

+ kgk



. (IV.3.33)

Let us now estimate the term A + B. For ﬁxed ϕ ∈ C

∞

), we have

I ≡ (D

+ B

), ϕ) ≡

ϕ(x)D



(x − y)[f

(y) − D

(y)]dy

−

−η

, x

(η

, 0)dη



ϕ(x)D



(x − y)[f

(y) −D

(y)]dy

−

, x

)



(η

− y

, y

)[f

(y)−D

(y)]dy



dη



and, therefore, after integration by parts, we arrive at

I =

(y) − D

(y)]D

dy (IV.3.34)

with

(y) =



(x − y) −

−η

, x

(η

−y

, y

)dη



ϕ(x)dx.

Denote by Z

(x, y) the function in curly brackets in this integral. It is easy

to show that, for every ﬁxed y ∈ R

−

, it holds that

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 259

(x, y) = 0, for a ll x ∈ R

. (IV.3.35)

Actually, for y ∈ R

−

, by w hat we already proved in this section and the

properties of the tensor U , both

(x − y)

and

−η

, x

(η

− y

, y

)dη

as functions of x solve the Stokes system in R

and assume the same value

at Σ. Moreover, they both have second deriva tives that are summable in R

to the qth power, 1 < q < ∞. Therefore, by Theorem IV.3.1, their diﬀerence

d(x) (say) can be at most a (suitabl e) linear function of x

. However, as is

immediately seen, d(x) tends to zero as x

tends to inﬁnity and (IV.3.35 ) is

therefore established. Setting

`ik

(y) ≡ D

(y),

from (IV.3.35) we obtain, in particular,

`ik

(y) = 0, for all y ∈ R

−

. (IV.3.36)

We shall show next that ζ

`ik

∈ D

1,q

) and

|ζ

`ik

1,q

≤ ckϕk

q,B

(IV.3.37)

for som e c independent of ρ. To this end, we observe that

`ik

= D

(x − y)ϕ(x)dx + D

(η

−y

, y

)χ

(η

, 0)dη

≡ ζ

(1)

+ ζ

(2)

(IV.3.38)

where

(η

, η

) =

−η

, x

− η

)ϕ(x)dx. (IV.3.39)

By the Calder´on–Zygmund theorem, we have

|ζ

(1)

1,q

≤ c

kϕk

q,B

(IV.3.40)

with c

= c

(q, n). Moreover, it is not diﬃcult to show the following two

statements:

(i) U

(η

−y

, y

) satisﬁes the assumptions made on the kernel k in Theorem

II.11.6;

(ii) χ

(η

, 0) ∈ L

(Σ).

260 IV Steady Stokes Flow in Bounded Domains

We may therefore apply Theorem II.11.6 to deduce

|ζ

(2)

1,q

≤ c

max

i,j

hhχ

(η

, 0)ii

1−1/q,q

(IV.3.41)

with c

= c

(q, n). Applying the trace Theorem II.10.2, from (IV.3.41) we

recover

|ζ

(2)

1,q

≤ c

max

i,j

|χ

1,q

. (IV.3.42)

However, employing Lemma II.11.1, it readily fol lows that for each ﬁxed m, i,

and j the kernels D

are li near combinations of kernels, each satisfying

the hypotheses of the Calder´on–Zygmund theorem, so that from (IV.3.39) it

follows that

|χ

1,q

≤ c

kϕk

q,B

with c

= c

(q, n). This inequality, together with (IV.3.42), furnishes

|ζ

(2)

1,q

≤ c

kϕk

q,B

which, along with (IV.3.40), proves (IV.3.37). In view of Theorem II.7.7, from

(IV.3.36) and (IV.3. 37) we conclude

`ik

∈ D

1,q

)

|ζ

`ik

1,q

≤ c

kϕk

q,B

(IV.3.43)

for a constant c

independent of ρ. With a view to (IV.3.43) and (IV.3.36),

from equation (IV.3.34) we derive

|I| ≡ |(D

+ B

), ϕ) ≤ c

−∇g

−1,q,R

kϕk

q,B

≤ c



|f |

−1,q,R

+ kgk

q,R



kϕk

q,B

so that, by the arbitrariness of ϕ and ρ, we obtain

|A + B|

1,q

≤ c

(| f|

−1,q

+ kgk

) . (IV.3.44)

From (IV.2.13), (IV.3.33), and (IV.3 .44) we then conclude

|v|

1,q

≤ c



|f |

−1,q

+ kgk

+ hhφii

1−1/q,q



By similar argument one shows

kpk

≤ c



|f|

−1,q

+ kgk

+ hhφii

1−1/q,q



and thus the existence proof is complete, at least for smooth f and g. If

f and g merely satisfy the assumptions formulated in the theorem, we can

easily establish existence by means of the estimate (IV.3.32) and the usual

density argument w hich, this time, makes use of Theorem II. 8.1. The proof

of uniqueness is entirely analogous to that given in Theorem IV.2.2 and it is

therefore omitted. The theorem is completely proved. ut

IV.3 Existence, Uniqueness, and L

-Estimates in a Half-Space 261

A simple, i nteresting consequence of Theorem IV.3.3 is the following.

Corollary IV.3.1 Given

g ∈ L

) ∩ L

), 1 < q, q

< ∞,

there exists v ∈ D

1,q

) ∩ D

1,q

) such that

∇ · v = g in R

|v|

1,r

≤ c k gk

, r = q, q

(IV.3.45)

where c = c(n, r).

In the last part of this section we shall provide the Green’s tensor (of the

ﬁrst kind) for the Stokes system in the half-space. We look for a tensor ﬁeld

G(x, y) = {G

(x, y)} and for a vector ﬁeld g(x, y) = {g

(x, y)} such that for

all j = 1, . . . , n:

∆

(x, y) +

∂g

(x, y)

∂x

= 0, x, y ∈ R

, x 6= y

∂G

(x, y)

∂x

= 0, x, y ∈ R

(x, y) = 0, x ∈ Σ ≡R

n−1

×{0}, y ∈ R

lim

|x|→∞

(x, y) = 0, y ∈ R

and, moreover, as |x − y| → 0

(x, y) = U

(x −y) + o(1), g

(x, y) = q

(x − y) + o(1),

where U , q is the Sto kes fundamental solution (IV.2.3), (IV.2.4). The pair

G, g is the Green’s tensor for the Stokes problem in the half-space and is the

vector counterpart of the Green’s function for the Laplace operator given in

(III.1.35). We can provide an explicit form of G and g, see, e.g., Maz’ja,

Plamenevski

i, & Stupyalis (1974 , Appendix 1), and one has for j = 1, . . ., n

and y

∗

= (y

, . . ., −y

)

(x, y) = U

(x − y) −U

(x − y

∗

) + W

(x, y), i = 1, . . .n − 1

(x, y) = U

(x − y) + U

(x − y

∗

) + W

(x, y),

(x, y) = q

(x − y) − q

(x − y

∗

) − t

(x, y), i = 1, . . .n − 1

(x, y) = q

(x −y) + q

(x − y

∗

) −t

(x, y)

(IV.3.46)

262 IV Steady Stokes Flow in Bounded Domains

where W

(x, y), t

(x, y) satisfy for all j = 1, . . . , n

(x, y) = −

− η

, x

)[U

(η

−y

, −y

)

−U

(η

−y

, y

)]dη

, i = 1, . . ., n − 1

(x, y) = −

− η

, x

)[U

(η

−y

, −y

)

−U

(η

− y

, y

)]dη

(x, y) = −D

k(x

−η

, x

)[U

(η

− y

, −y

)

−U

(η

−y

, y

)]dη

, i = 1, . . ., n − 1

(x, y) = −D

k(x

−η

, x

)[U

(η

− y

, −y

)

−U

(η

− y

, y

)]dη

(IV.3.47)

where the kernels K

and k are deﬁned in (IV.3.4). In particular, one ﬁnds

for n = 3:

(x, y) =

4π

∂

∂x

∂y



|x − y

∗



, i, j = 1, 2,

(x, y) = −

4π

∂

∂x



|x −y

∗



4π

∂

∂x

∂y



|x −y

∗



, j = 1, 2 ,

(x, y) = −

4π

∂

∂y



|x −y

∗



4π

∂

∂x

∂y



|x − y

∗



, i = 1 , 2,

(x, y) =

4π|x − y

∗

−

4π

∂

∂x



|x −y

∗



−

4π

∂

∂y



|x −y

∗



4π

∂

∂x

∂y



|x −y

∗



(x, y) =

2π

∂

∂x

∂y



|x −y

∗



, i = 1, 2,

(x, y) = −

2π

∂

∂x



|x −y

∗



2π

∂

∂x

∂y



|x −y

∗



(IV.3.48)

(x −y

∗

) is regular for all x, y ∈ R

. Notice that, unlike the analogous Green’s

function for the Laplace operator, the function

(x −y) − U

(x −y∗)

cannot be taken as the Green tensor for the Stokes problem, since the solenoidality

condition is not satisﬁed. Thus, we must modify it by adding functions W

IV.4 Interior L

-Estimates 263

while, for n = 2:

(x, y) =

2π

∂

∂x

∂y



|x −y

∗



(x, y) = −

2π

∂

∂x



|x − y

∗



4π

∂

∂x

∂y



|x − y

∗



(x, y) = −

2π

∂

∂y



|x −y

∗



2π

∂

∂x

∂y



|x −y

∗



(x, y) =

2π

|x −y

∗

−

2π

∂

∂x



|x − y

∗



−

2π

∂

∂y



|x −y

∗



2π

∂

∂x

∂y



|x −y

∗



(x, y) = −

2π

∂

∂x

∂y



|x − y

∗



(x, y) = −

2π

∂

∂x

∂y



|x − y

∗



(IV.3.49)

From (IV.3.46)–(IV.3.49) we wish to single out some estimates for G, g that

will be useful later. Precisely, by a simple computation, we ﬁnd for n = 3,

(x, y) ≤ |x − y|

−1−|α|

(x, y) ≤ |x − y|

−2−|α|

(IV.3.50)

and for n = 2,

(x, y)| + |D

(x, y)| ≤ c|x − y|

−1−|α|

, (IV.3.51)

where |α| ≥ 0, c = c(n) and D

, D

are acting either on x or y.

Exercise IV.3.3 Starting from (IV.3.46), (IV.3.47), prove the following estimates

for all n ≥ 3 and all |α| ≥ 0

(x, y)| ≤ c|x − y|

−n−|α|+2

(x, y)| + |D

(x, y)| ≤ c|x − y|

−n−|α|+1

(IV.3.52)

IV.4 Interior L

-Esti mates

In this section we will investigate the properties of generalized solutions “far”

from the boundary of the region of motion by means of the results established

264 IV Steady Stokes Flow in Bounded Domains

in Section IV.2 for solutions in the whole space. Actually, we consider a pair

of functions v, p with v ∈ W

1,1

loc

(Ω), ∇ · v = 0 in the generalized sense and

p ∈ L

loc

(Ω) satisfying identity (IV.1.3) for all ψ ∈ C

∞

(Ω) and shall show that,

for suitable f , the ﬁelds v, p obey certain L

-inequalities that imply, among

other things, that any weak solution is in fact of class C

∞

(Ω), provided f

enjoys the same property.

We begin to prove some preliminary results. First of all, the regularizations,

, p

, of v and p, respectively, obey the Sto kes equation in any subdomain

Ω

with Ω

⊂ Ω. This result is a straightforward consequence of the following

very general one.

Lemma IV.4.1 Let Ω be an arbitrary domain of R

, n ≥ 2. Suppose that

v ∈ L

loc

(Ω) obeys the conditions

(v, ∆ϕ) = hf , ϕi, for all ϕ ∈ D(Ω) ,

(v, ∇ψ) = 0 , for all ψ ∈ C

∞

(Ω),

(IV.4.1)

where f satisﬁes either (i) f ∈ L

loc

(Ω), or (ii) f ∈ D

−1,q

(ω), for all ω with

ω ⊂ Ω, and some q > 1. Then, for any domain Ω

with Ω

⊂ Ω, we have

∆v

= ∇p

(ε)

+ g

∇ · v

= 0

)

in Ω

, (IV.4.2)

for some p

(ε)

∈ C

∞

(Ω), and where g = f , in case (i), while, i n case (ii),

= ∇ ·F

, where F ∈ L

(Ω

) satisﬁes hf , χi = (F , ∇χ), for all χ ∈ D(Ω

)

(see Theorem II.1.6 and Theorem II.8.2).

Proof. Let ϕ ∈ D(Ω

). Then ϕ

∈ D(Ω), whenever ε < dist (Ω

, ∂Ω), and can

be thus replaced in (IV.4.1)

. Using this latter relation, by a straightforward

calculation that uses Exercise II.3. 2, we then show

(∆v

− g

, ϕ) = 0 , for al l ϕ ∈ D(Ω

) .

Therefore, (IV.4 .2)

follows from this equation and from Lemma III.1.1. The

second equation in (IV.4 .2) can be proved from (IV.4.1) in exactly the same

way. ut

The previo us result trivially im plies the following one.

Lemma IV.4.2 Let Ω be an arbitrary domain of R

, n ≥ 2, and let v, p with

v ∈ W

1,1

loc

(Ω), ∇ · v = 0, and p ∈ L

loc

(Ω) satisfy (IV.1.3) for all ψ ∈ C

∞

(Ω)

and with f ∈ L

loc

(Ω). Then the regularizations, v

, p

∆v

= ∇p

+ f

∇ · v

= 0

)

in Ω

, (IV.4.3)

for a ll doma ins Ω

with Ω

⊂ Ω.

IV.4 Interior L

-Estimates 265

Employing Theorem IV.2.1 and Lemma IV.4.2 we shall show the following

interior regularity result.

Theorem IV.4.1 Let Ω be an arbitrary domai n in R

, n ≥ 2. Let v be

weakly divergence-free with ∇v ∈ L

loc

(Ω), 1 < q < ∞,

and satisfying

(IV.1.2) for all ϕ ∈ D(Ω). Then, if

f ∈ W

m,q

loc

(Ω), m ≥ 0,

it follows that

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω)

where p is the pressure ﬁeld associated to v by Lemma I V.1.1. Further, the

following inequality holds:

|v|

m+2,q,Ω

+ |p|

m+1,q,Ω

≤ c (kfk

m,q,Ω

+ kvk

1,q,Ω

−Ω

+ kpk

q,Ω

−Ω

)

(IV.4.4)

where Ω

, Ω

are arbitrary bounded subdomains of Ω with Ω

⊂ Ω

, Ω

⊂ Ω,

and c = c(n, q, m, Ω

, Ω

Proof. Consider a “cut-oﬀ” function ϕ ∈ C

∞

) that is one in Ω

and zero

outside Ω

Choosing in (IV.4.4) Ω

⊃ Ω

and multiplying (IV.4.3) by ϕ,

after a simple manipulation we obtain that the functions

u = ϕv

, π = ϕp

satisfy

∆u = ∇π + f

+ f

∇ · u = g,

(IV.4.5)

where

= ϕf

, f

= −p

∇ϕ + 2∇ϕ · ∇v

+ v

∆ϕ

g = ∇ϕ · v

(IV.4.6)

Problem (IV.4.5) can be considered in the whole of R

, by extending v

, p

and f

to zero outside Ω

. Since D

u ∈ L

) we may apply Theorem

IV.2. 1 with m = 0 to deduce the foll owing estimate

+ k∇πk

≤ c

(kϕf

+ k∇(∇ϕ ·v

k∇ϕ · ∇v

+ kv

∆ϕk

+ kp

∇ϕk

) ,

(IV.4.7)

where c

= c

(n, q). From (IV.4.6), (IV.4.7) and from the properties of ϕ we

obtain

Notice that, by Lemma II.6.1, v ∈ L

loc

(Ω). The assumption on v can be weak-

ened. We shall show this directly for the nonlinear case in Chapter IX.

For instance, we may choose ϕ as the regularization (χ

Ω

)

of the characteristic

function of the domain Ω

and take ε suﬃciently small.