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

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236 IV Steady Stokes Flow in Bounded Domains

p ∈ L

(Ω).

Finally, if we normalize p by the condition

Ω

p = 0, (IV.1.4)

the following estimate holds:

kpk

≤ c (kfk

−1,q

+ |v|

1,q

) . (IV.1.5)

Proof. We b egin to prove the ﬁrst part. It is enough to show that (IV.1.2)

implies (IV.1 .3), the reverse implication being obvious. Let us consider the

functional

F(ψ) ≡ (∇v, ∇ψ) + hf , ψi

for ψ ∈ D

1,q

(Ω

). By assumption, F is bounded in D

1,q

(Ω

) and is identically

zero in D(Ω) and, therefore, by continuity, in D

1,q

(Ω

). If Ω is arbitrary (in

particular, has no regularity), from Corollary III.5.2 we deduce the existence

of p ∈ L

loc

(Ω) verifying (IV.1.3) for all ψ ∈ C

∞

(Ω). If Ω is bounded and

satisﬁes the cone condition, by assumption and Corollary III.5.1 there exists

a uniquely determined p

∈ L

(Ω) with

Ω

= 0

such that

F(ψ) = (p

, ∇· ψ), (IV.1.6)

for a ll ψ ∈ D

1,q

(Ω). From (IV.1.3) and (IV.1.6) we ﬁnd, in particular,

(p − p

, ∇ · ψ) = 0, for all ψ ∈ C

∞

(Ω),

implying p = p

+const. (see Exercise II.5.9), and so, if we normalize p by

(IV.1.4) we may take p = p

. Consider the problem

∇· ψ = |p|

q−2

p −

|Ω|

Ω

|p|

q−2

p ≡ g

ψ ∈ W

1,q

(Ω)

kψk

1,q

≤ c

kpk

q−1

(IV.1.7)

with Ω bounded and satisfying the cone condition. Since

Ω

g = 0, g ∈ L

(Ω), kgk

≤ c

kpk

q−1

from Theorem III.3.1 we deduce the existence o f ψ solving (IV.1.7). If we

replace such a ψ into (IV.1.6) and use (IV.1.4) together with the H¨o lder

inequality and inequality (II.3.22)

, we obtain (IV.1.5). The proof i s therefore

completed. ut

IV.1 Generalized Solutions. Existence and Uniqueness 237

Remark IV.1.3 If we relax the normalizatio n condition (IV.1. 4) on p, in

place of (IV.1.5) one can show, as the reader will easily check, the inequality

inf

c∈R

kp + ck

≤ c (kfk

−1,q

+ |v|

1,q

) .



We now pass to the pro of of existence and uniqueness of weak solutions.

Theorem IV.1.1 Let Ω ⊆ R

, n ≥ 2, be bounded and locally Lipschitz. For

any f ∈ D

−1,2

(Ω) and v

∗

∈ W

1/2,2

(∂Ω) verifying

∂Ω

∗

· n = 0,

there exists one and only one weak solution v to the Stokes problem (IV.0.1),

(IV.0.2). Moreover, if we denote by p the correspo nding pressure ﬁeld associ-

ated to v by L emma IV.1.1, the following estimate holds:

kvk

1,2

+ kpk

≤ c



kfk

−1,2

+ kv

∗

1/2,2(∂Ω)



, (IV.1.8)

where c = c(n, Ω).

Proof. By the results of Exercise III.3.8 there exists a solenoidal extension

V ∈ W

1,2

(Ω) of v

∗

such that

kV k

1,2

≤ c

∗

1/2,2(∂Ω)

(IV.1.9)

with c

independent of V and v

∗

. We look for a generalized solution of the

form v = w + V where w ∈ D

1,2

(Ω) satisﬁes the identity

(∇w, ∇ϕ) = −hf, ϕi − (∇V , ∇ϕ), (IV.1.10)

for all ϕ ∈ D

1,2

(Ω). The right-hand side of (IV.1.10) deﬁnes a bounded linear

functional in D

1,2

(Ω) and so, by the Riesz representation theorem, there exists

one and only one w ∈ D

1,2

(Ω) verifying (IV.1.10). This shows existence of

a weak solution. To prove uniq ueness, denote by v

another weak solution

corresponding to the same data. Evidently, Theorem II.4.2 furnishes that u ≡

v −v

is an element of

1,2

(Ω) and, therefore, by the results of Section III.5,

of D

1,2

(Ω). On the other hand by (iv) of Deﬁnition IV.1.1 it follows that

(∇u, ∇ϕ) = 0

for all ϕ ∈ D

1,2

(Ω), implying u = 0 a.e. in Ω. To show estima te (IV.1.8),

we ta ke ϕ = w into (IV.1.10), appl y the Schwarz inequality and inequality

(II.3.22)

, and use (IV.1.9) and (II.5.1) together with (III.3.14) to obtain for

some c

= c

(n, Ω)

kwk

1,2

≤ c



kf k

−1,2

+ kv

∗

1/2,2(∂Ω)



. (IV.1.11)

Estimate (IV.1. 8) then follows from (IV.1.4), (IV.1.9), and (IV.1.11). ut

238 IV Steady Stokes Flow in Bounded Domains

Remark IV.1.4 If v

∗

≡ 0, the existence of a generalized solution is estab-

lished without regulari ty assumptions on Ω. 

Exercise IV.1.1 Theorem IV. 1.1 also remains valid if ∇ · v = g 6≡ 0, where g is

a suitably ascribed f unction. Speciﬁcally, show that for Ω, f , and v

∗

satisfying the

assumption of that theorem and all g ∈ L

(Ω) such that

Ω

g =

∂Ω

∗

· n

there exists one and only one weak solution v to the nonhomogeneous Stokes prob-

lem, that is, a ﬁeld v : Ω → R

satisfying (i), (iii), and (iv) of Deﬁnition IV.1.1,

with q = 2, and ∇·v = g (weakly). Show, in addition, that v and the corresponding

pressure ﬁeld p obey the following esti mate:

kvk

1,2

+ kpk

≤ c

kf k

−1,2

+ kv

∗

1/2,2(∂Ω)

+ kgk

Hint: Look for a solution of the form v = w + V where w veriﬁes (IV.1.7), while V

solves ∇·V = g, in Ω, V = v

∗

at ∂Ω and use the results of Exercise III. 3.8.

IV.2 Existence, Uniqueness, and L

-Esti mates in the

Whole Space. The Stokes Fundamental Solution

Our next task is to establish interior and boundary inequalities for solutions to

the Stokes problem that will furnish, in particular, that generalized solutions

are in fact classical if the domain and data are suﬃciently smooth. We ﬁrst

derive these estimates in two special cases, namely, when either Ω = R

Ω = R

. The job here is easier because we are able to furnish solutions

of explicit form. To this end, let us introduce the fundamental solution for

the Stokes equation (IV.0.1), which plays the same role as the fundamental

solution of Laplace’s equation.

Consider the second-order, symmetric tensor

ﬁeld U and the vector ﬁeld q deﬁned by the relations

(x − y) =



∆ −

∂

∂y



Φ(|x −y|)

(x − y) = −

∂

∂y

∆Φ(x − y),

(IV.2.1)

where x, y ∈ R

, δ

is the Kronecker symbol and Φ(t) is an arbitrary function

on R, which is smooth for t 6= 0. Noticing that ∂|x −y|/∂x

= −∂|x −y|/∂y

by a simple calculation from (IV.2.1) one has for x 6= y and all i, j = 1, . . ., n

Actually, all the material presented in this and in the subsequent section will

be derived along the same lines of the one developed for the Dirichlet problem

for the Poisson equation at the end of Section II.7 (see Exercise II.11.9, Exercise

II.11.10, and Exercise II.11.11.

We recall that, according to Einstein’s convention, unless otherwise explicitly

stated, pairs of identical indices imply summation from 1 to n.

IV.2 Existence, Uniqueness, and L

-Estimates in the Whole Space 239

∆U

(x − y) +

∂

∂x

(x − y) = δ

∆

Φ(x − y)

∂

∂x

(x − y) = 0.

(IV.2.2)

Choose now Φ as the fundamental solution to the biharmonic equation. So,

for n = 3,

Φ(|x − y|) = −

|x − y|

8π

and the associated ﬁelds U and q become (Lo rentz 1896)

(x − y) = −

8π



|x − y|

−y

)(x

− y

)

|x −y|



(x − y) =

4π

− y

|x − y|

(IV.2.3)

Likewise, for n = 2 ,

Φ(|x − y|) = |x − y|

log(|x −y|)/8π

and we have

(x − y) = −

4π



log

|x −y|

− y

)(x

−y

)

|x − y|



(x − y) =

2π

−y

|x −y|

(IV.2.4)

Moreover, with the above choice of Φ, from (IV.2.2 ) it follows that the ﬁelds

(IV.2.3) and (IV.2.4) satisfy

∆U

(x − y) +

∂

∂x

(x − y) = 0

∂

∂x

(x − y) = 0.

for x 6= y. (IV.2.5)

The pair U , q is called the fundamental solution of t he Stokes equation.

Remark IV.2.1 In dimension n > 3 the fundamental solution is given by

(IV.2.1) with

Φ =







−(1/8π

) log |x −y| if n = 4



Γ (n/2 −2)/16π

n/2



|x − y|

4−n

if n ≥ 4.

One thus has for all n ≥ 4

240 IV Steady Stokes Flow in Bounded Domains

(x − y) = −

2n(n − 2)ω



|x − y|

n−2

+ (n − 2)

−y

)(x

−y

)

|x −y|



(x − y) =

nω

−y

|x −y|



From (IV.2.3) and (IV.2.4) (and Remark IV.2.1 ), we may formally compute

the asymptotic properties of U and q. In particular, the following estimates,

as either |x| → 0 or |x| → ∞, are readily established:

U(x) = O(l og |x|) if n = 2,

U(x) = O(|x|

−n+2

) if n > 2,

U(x) = O(|x|

−n−|α|+2

), |α| ≥ 1, n ≥ 2,

q(x) = O(|x|

−n−|α|+1

), |α| ≥ 0, n ≥ 2.

(IV.2.6)

Let us now consider the foll owing nonhomogeneous Stokes problem

∆v = ∇p + f

∇ · v = g

)

in R

, (IV.2.7)

where f and g are prescribed functions from C

∞

). Using (IV.2.3 ) and

(IV.2.4) it is not diﬃcult to prove the exi stence of solutions to (IV.2.7), veri-

fying suitable L

-estimates. To reach this goal, we introduce the Stokes volume

potentials

u(x) =

U(x − y) · F (y)dy

π(x) = −

q(x − y) · F (y)dy,

(IV.2.8)

where F ∈ C

∞

). Since

U(x − y) · F (y)dy =

U(z) · F (x − z)dz

q(x − y) · F (y)dy =

q(z) · F (x − z)dz

one has u, π ∈ C

∞

). Moreover, it is easy to show that u, π is a solution

to (IV.2.7) with g ≡ 0 and f ≡ F . Actually, it is obvious that ∇·u = 0; also,

using (IV.2.1) and Exercise II.11.3, we deduce

∆u(x) − ∇π(x) = ∆

∆Φ(|x − y|)F (y)dy

= ∆(E ∗ F )(x) = F (x).

(IV.2.9)

IV.2 Existence, Uniqueness, and L

-Estimates in the Whole Space 241

We shall now look for a solution v, p to (IV.2.7) of the form v = u + h, p = π

where u and π a re volume potentials corresponding to F ≡ f − ∆h and

h = ∇(E ∗ g). (IV.2.10)

Since ∆h = ∇g ∈ C

∞

) and

∇· h = g, (IV.2. 11)

from (IV.2.9) and (IV.2.11) we may conclude that v, p is a solution to (IV.2.7).

Furthermore, from (IV.2.6) one shows as |x| → ∞

v(x) = O(l og |x|) if n = 2,

v(x) = O(|x|

−n+2

) if n > 2,

v(x) = O(|x|

−n−|α|+2

), |α| ≥ 1, n ≥ 2,

p(x) = O(|x|

−n−|α|+1

), |α| ≥ 0, n ≥ 2.

(IV.2.12)

Let us now derive some L

-inequalities fo r v, p in terms of g and f . From

(IV.2.10) and the Calder´on–Zygmund Theorem II .7.4 we have

|h|

`+1,q

≤ c|g|

`,q,

for a ll ` ≥ 0 (IV.2.13)

with c = c(n, q). Next, consider the identity with |α| = `

(x) =

(x − y)D

(y)dy

= lim

ε→0

|x−y|≥ε

(x − yD

(y)dy

+lim

ε→0

|x−y|=ε

(x − yD

(y)n

(y)dσ

(IV.2.14)

where n

is the jth component of the unit outer normal to the sphere |x −y| =

ε. From (IV.2.3) and (IV.2.4) one has that D

is ho mogeneous of degree

1 − n, so that by Lemma II.11.1, D

is a singular kernel. Furthermore,

again by that lemma, it follows that

lim

ε→0

|x−y|=ε

(x − y)D

(y)n

(y)dσ

= A

ijk`

(x)

with A

ijk`

a constant fourth-order tensor. Combining this formula with

(IV.2.14) gives

More detailed estimates wi ll be given in Section V.3.

242 IV Steady Stokes Flow in Bounded Domains

(x) = lim

ε→0

|x−y|≥ε

(x − y)D

(y)dy + A

ijk`

(x),

where the integral is to be understood in the Cauchy principal value sense.

We may now employ in this i dentity the Calder´on–Zygmund theorem and

(IV.2.13) to obtain for all ` ≥ 0 a nd all q > 1

|u|

`+2,q

≤ c

(| f|

`,q

+ |g|

`+1,q

) , (IV.2.15)

where c

= c

(n, q). Likewise, one proves

|π|

`+1,q

≤ c

(|f |

`,q

+ |g|

`+1,q

) . (IV.2.16)

From (IV. 2.13), (IV.2.15), and (IV.2.16) we thus obtain the following estimate

for the solution v, p, valid for all ` ≥ 0 and all q > 1

|v|

`+2,q

+ |p|

`+1,q

≤ c (|f|

`,q

+ |g|

`+1,q

) (IV. 2.17)

with c = c(n, q).

Other estimates can be obtained directly from (IV.2.8) and (IV.2.10), by

noting that

u(x)| + |D

π(x)| ≤ c

F (y)|

|x −y|

n−1

h(x)| ≤ c

g(y)|

|x −y|

n−1

dy.

If 1 < q < n, we may thus apply the Sobolev Theorem II.7.3 to obtain

|v|

`+1,s

+ |p|

`,s

≤ c

(| f|

`,q

+ |g|

`+1,q

) , s

n − q

. (IV.2. 18)

Likewise, if 1 < q < n/2, from (IV.2.18) and (II.6.17) we have

|v|

`,s

≤ c

(|f|

`,q

+ |g|

`+1,q

) , s

n − 2q

. (IV.2.19)

Assume now f and g m erely belonging to W

m,q

) and D

m+1,q

respectively, m ≥ 0 and q ∈ (1, ∞). We can approximate them with sequences

}, {g

} ⊂ C

∞

). Denoting by {v

, p

} the corresponding sequence

of solutions to (IV.2.7), we see that each solution satisﬁes (IV.2.17) for all

` ∈ [0, m] and, if 1 < q < n [respectively, 1 < q < n/2], it satisﬁes al so

(IV.2.18) [respectively, (IV.2.19)]. Employing these estimates together with

the weak compactness property of spaces

m,q

(see Exercise II.6.2), one easily

shows the existence of two ﬁelds v and p such that

v ∈ B ≡

`=0

`+2,q

), p ∈ P ≡

`=0

`+1,q

)

IV.2 Existence, Uniqueness, and L

-Estimates in the Whole Space 243

and

lim

k→∞

, ψ) = (D

v, ψ) , 0 ≤ |α| ≤ m + 2

lim

k→∞



∇p

, ψ





∇p, ψ



0 ≤ |β| ≤ m

for all ψ ∈ L

). This implies, in particular, that the pair v, p satisﬁes

(IV.2.7) a.e. in R

along with estima tes (IV.2.17)–(IV. 2.19). Furthermore, by

Lemma I I.6.1, we have

v ∈ W

m+2,q

), p ∈ W

m+1,q

for a ll R > 0.

Let now v

, p

denote another solution to (IV.2.7) correspo nding to the

same data as v, p, with |v

`+2,q

ﬁnite, for some ` ∈ [0, m]. It is then easy to

show that |v

− v|

`+2,q

= |p

− p|

`+1,q

= 0.

In fact, setting z = v

− v and

τ = p

− p, we obtain

∆z = ∇τ

∇ · z = 0

(IV.2.20)

a.e. in R

. It follows at once tha t

∇τ · ∇ψ = 0

for any ψ ∈ C

∞

). Since D

∇τ ∈ L

), |α| = `, by a well-known result of

Caccioppoli (1937), Cimmino (1938a, 1938b), and Weyl (1940) we then deduce

that τ is harmonic, and hence smooth, in the who le space. As a consequence,

by Exercise I I.11.11 it follows that D

∇τ = 0. Therefore, (IV.2.2 0)

furnishes

∆D

z = 0 and, again by Exercise II.11 .11, we have |z|

`+2,q

= 0, which is

what we wanted to prove.

We collect the results obtained so f ar in the following.

Theorem IV.2.1 Given

f ∈ W

m,q

), g ∈ D

m+1,q

), m ≥ 0, 1 < q < ∞, n ≥ 2,

there exists a pair of functions v, p such that v ∈ W

m+2,q

), p ∈

m+1,q

) for any R > 0, satisfying a.e. the no nhomogeneous Stokes sys-

tem (IV.2.7). Moreover, for all ` ∈ [ 0, m], |v|

`+2,q

and |p|

`+1,q

are ﬁnite and

we have

|v|

`+2,q

+ |p|

`+1,q

≤ c (|f|

`,q

+ |g|

`+1,q

) . (IV.2.21)

If, in particular, n/2 ≤ q < n, then |v|

`+1,s

and |p|

`,s

, s

= nq/(n − q), are

ﬁnite, and we have

Notice that if v is a solution to ( IV.2.7) having |v|

`+2,q

ﬁnite, then |p|

`+1,q

ﬁnite too.

244 IV Steady Stokes Flow in Bounded Domains

|v|

`+1,s

+ |p|

`,s

+ |v|

`+2,q

+ |p|

`+1,q

≤ c (| f |

`,q

+ |g|

`+1,q

) . (IV.2.22)

Furthermore, if 1 < q < n/2, then |v|

`,s

, s

= nq/(n − 2q), is ﬁnite and the

following inequality holds

|v|

`,s

+|v|

`+1,s

+|p|

`,s

+|v|

`+2,q

+|p|

`+1,q

≤ c (|f|

`,q

+ |g|

`+1,q

) . (IV.2.23)

In the a bove inequalities, c = c(n, q, `). In addition, if f, g ∈ C

∞

), then

v, p ∈ C

∞

) and they have for large |x| the asymptotic behavior indicated

in (IV.2.12). Finally, if v

, p

is another solution to (IV.2.6) corresponding to

the data f , g with |v

`+2,q

ﬁnite fo r some ` ∈ [0, m], then |v

− v|

`+2,q

= 0

and |p

− p|

`+1,q

= 0.

The last part of this section is devoted to show existence and uniqueness

of q-weak solutions to (IV.2.7). The results we obtain are similar to those of

Theorem IV.1.1 and Exercise IV.1.1, with the diﬀerence that now we consider

the problem in the general context of spaces D

1,q

, 1 < q < ∞.

To this end, we give the following

Deﬁnition IV.2.1 A vector ﬁeld v is a q-generalized solution to (IV.2.7) if

and only if

(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

∞

Lemma IV.1.1 implies the following result.

Lemma IV.2.1 Let f ∈ W

−1,q

), for all R > 0. Then, to every q-

generalized solution in the sense of Deﬁnition IV.1.2, we may associate a

pressure ﬁeld p ∈ L

), all R > 0, such that

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

∞

). (IV.2 .24)

If, in particular, f ∈ D

−1,q

), then p ∈ L

) , and the fol lowing estimate

holds

kpk

≤ c (|v|

1,q

+ kfk

−1,q

) .

We then have

Theorem IV.2.2 Given

f ∈ D

−1,q

), g ∈ L

), 1 < q < ∞ , n ≥ 2,

there exists at least one q-generalized solution, v to (IV.2.7). Moreover, de-

noting by p the pressure ﬁeld asso ciated to v by Lemma IV.2.1 , we have

|v|

1,q

+ kpk

≤ c (|f|

−1,q

+ kgk

) . (IV.2 .25)

IV.2 Existence, Uniqueness, and L

-Estimates in the Whole Space 245

If q ∈ (1, n), then v ∈ L

nq/(n−q)

) and the following inequality holds

kvk

nq/(n−q)

+ |v|

1,q

+ kpk

≤ c (| f |

−1,q

+ kgk

) . (IV.2.26)

Finally, if v

is a q

-generalized solution (1 < q

< ∞, q

possibly diﬀerent

from q) corresponding to the same f and g, it fol lows that v

= v + c

a.e.

in R

, for some constant vector c

, with c

= 0 if q < n and q

< n, and,

denoting by p

the pressure ﬁeld associated to v

by Lemma IV.2.1, we have

also p

= p + const. a.e. in R

Proof. We begin to observe that if we prove the existence of a q-generalized

solution satisfying (IV.2.25), then the validity of (IV. 2.26) follows from this

equation and Theorem II.7.6. We next notice that it is enough to show the

existence result for f , g ∈ C

∞

) (f

satisfying (II.8.10) when q

≥ n, for

all i = 1, . . ., n). The general case will then follow by a standard density ar-

gument that uses (IV.2.2 5), the weak compactness property of spaces D

1,q

(see Exercise II.6.2), Theorem II.8.1 and the density of C

∞

) in L

Actually, given f ∈ D

−1,q

), g ∈ L

), let {f

}, {g

} ⊂ C

∞

) be two

sequences approximating f and g. If existence of a solution {v

, p

} is estab-

lished for each f

and g

, by (IV.2.25) and the weak compactness property of

1,q

and L

, 1 < q < ∞ (see Exercise II.6.2 and Theorem I I.2.4(ii )), we may

ﬁnd two ﬁelds v ∈ D

1,q

) and p ∈ L

) such that, for a ll φ ∈ L

(Ω),

lim

k→∞

)

, φ) = (D

, φ), lim

k→∞

, φ) = (p, φ), i, j = 1, . . .n,

and which, by Theorem II.2.4(i), obey (IV.2.25). Furthermore, since for any

k ∈ N

(∇v

, ∇ψ) − (p

, ∇ · ψ) = −[f

, ψ], for all ψ ∈ C

∞

we take the lim it k → ∞ into this identity and use the density prop erties of

∞

into D

−1,q

and L

, thus proving existence in the general case. Therefore,

we need to show existence for smooth f and g only. In such a case, we k now

that a solution to the problem is gi ven by v = v

+ v

+ h, p = p

+ p

where h is deﬁned in (IV.2.10) and v

= U ∗ f , v

= U ∗ ∆h, p

= −q ∗ f ,

= −q ∗∆h. From (IV.2.13 ) we obtain

1,q

+ |h|

1,q

≤ c

kgk

(IV.2.27)

with c

= c

(n, q). On the other hand, for ﬁxed ρ > 0 and arbitrary ϕ ∈

) we have (extending ϕ to zero in B

)

q,B

= sup

kϕk

≤1

|(D

)

, ϕ)|

= sup

kϕk

≤1



(y)



(x − y)ϕ(x)dx





(IV.2.28)