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

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856 XII Two-Dimensional Flow in Exterior Domains

the ﬁelds given in (XII.2.7) for R ∈ (1, 2) provide examples of generalized

solutions that decay more slowly than any negative power of r.

In the current section we shall prove some prelim inary results based on

the work of Galdi and Sohr (1995) and Sazonov (1999). In particular, we

deduce that every generalized solution corresponding to v

∞

≡ (1, 0)

has

the same global summability properties of the Oseen fundamental tensor,

provided f is of bounded support.

This will imply, in particular, that the

velocity ﬁeld and the associated pressure of these generalized solutions satisfy

the energy equality. Moreover, we obtain representation formulas identical to

those derived in Section X.5 for the three-dimensional case.

We begin by showing some existence and uniqueness results for a suitable

linearization of (XII.0.1), (XII.0.2). Speciﬁcally, let us consider the following

problem

∆u +

∂u

∂x

= a

∂u

∂x

+ Au

+ ∇π + G

∇ · u = g,

(XII.7.1 )

where a, A, G and g are prescribed functions.

Lemma XII.7.1 Let

G ∈ L

), g ∈ W

1,q

), q ∈ (1, 3/2)

A ∈ L

), a ∈ L

∞

Moreover, let u, π be any solution to (XII.7.1) such that for some q ∈ (1, 2),

∈ L

2q/(2−q)

), D

∂u

∂x

∈ L

and

lim

|x|→∞

(x) = 0.

Then, there exists a posi tive constant c = c(q, q) (cf. (XII.7.4) a nd (XII.7.11))

such that if

kak

∞

+ kAk

< c,

we have for some π

∈ R

u ∈ D

2,q

) ∩ D

1,3q/(3−q)

) ∩ L

3q/(3−2q)

∈ D

1,q

) ∩L

2q/(2−q)

)

(π −π

) ∈ D

1,q

) ∩ L

2q/(2−q)

Of course, this choice causes no loss of generality whenever v

∞

6= 0.

More generally, we may assume f to satisfy suitable decay properties in a neigh-

borhood of inﬁnity.

XII.7 Global Summability of Generalized Solutions when v

∞

6= 0 857

Proof. Let X

, 1 < q < 3/2, denote the Banach space of solenoida l functions

w ∈ L

loc

) such that the norm

kwk

:= kw

2q/(2−q)

+ k∇w



∂w

∂x



+ k∇wk

3q/(3−q)

+kD

+ kwk

3q/(3−2q)

is ﬁnite. Denote by B

(δ)

the ball in X

of radius δ (> 0) and consider the map

L : w

∈ B

(δ)

→ w ∈ X

where w solves the problem

∆w +

∂w

∂x

= a

∂w

∂x

+ Aw

+ ∇τ + G

∇ · w = g.

(XII.7.2 )

Since for all q ∈ (1, 2), by the H¨ol der i nequality, we have



∂w

∂x

+ Aw



≤ kak

∞



∂w

∂x



+ kAk

2q/(2−q)

, (XII.7.3)

by the hypotheses made on G and g, the map L is well deﬁned for all q ∈

(1, 3/2). Furthermore, by Theorem VII.4.1 and (XII.7.3), fo r all w

∈ B

(δ)

ﬁnd

kwk

≤ c



(kak

∞

+ kAk

)kw

+ kGk

+ kgk

1,q



for som e c

= c

(q). Thus, assuming (for instance)

kak

∞

+ kAk

, (XII.7.4)

and choosi ng δ ≥ 2c

(kGk

+ kgk

1,q

), it follows that L transforms B

(δ)

into

itself. Moreover, from (XII.7.2) with G ≡ g ≡ 0, a nd by (XII.7.3), (XII.7.4)

we obtain

kwk

≤

and so the existence of a solutio n w, τ to (XII.7.1) with w ∈ X

follows from

the contraction mapping theorem. Moreover, τ ∈ D

1,q

) and, therefore, by

Theorem II.6.1 , τ − τ

∈ L

2q/(2−q)

), for some τ

∈ R. We shall now show

that u = w, τ = π+const a.e. in R

. To this end, letting

w = w − u, s = τ − π,

it follows that

∆w +

∂w

∂x

= a

∂w

∂x

+ Aw

+ ∇s

∇· w = 0.

(XII.7.5 )

858 XII Two-Dimensional Flow in Exterior Domains

It is easy to show that

∈ L

2q/(2−q)

∂w

∂x

∈ L

To this end, it is enough to prove that

∈ L

2q/(2−q)

∂w

∂x

∈ L

). (XII. 7.6)

Assume q < q (the other case q > q can be treated likewise by interchanging

the role of q and q). Since w ∈ D

2,q

), q < 2, by Theorem II.6 .1 we have

∈ D

1,2q/(2−q)

) (XII.7.7)

and so, since w

∈ L

2q/(2−q)

), from this, from (XII.7.7), and from Theorem

II.9.1 we obtain

∈ L

∞

which proves the ﬁrst relation in (XII.7.6). Furthermore, by the properties

w ∈ D

2,q

∂w

∂x

∈ L

and Lemm a II.3.3 we ﬁnd that

∂w

∂x

∈ L

), for a ll s ∈ [q, 2]

and also the second relation in (XII.7.6) follows. From (XII.7.4) with w

re-

placed by w we may thus conclude that

F ≡ a

∂w

∂x

+ Aw

∈ L

Therefore, i n view of Theorem VII.4.1, the problem

∆z +

∂z

∂x

= F + ∇σ

∇ · z = 0

admi ts at least one solution w

∗

, s

∗

, such that

∗

+ kw

∗

2q/(2−q)



∂w

∗

∂x



≤ c

kF k

≤ c

(kak

∞

+ kAk

)



∂w

∂x



+ kw

/(2−

)

(XII.7.8 )

with c

= c

(q). We shall now show that

XII.7 Global Summability of Generalized Solutions when v

∞

6= 0 859

∗

− w) ≡

∂(w

∗

− w)

∂x

≡ 0, w

∗

≡ w

. (XII.7.9)

Actually, setting v = w

∗

−w, p = s

∗

− s, it follows that

∆v +

∂v

∂x

= ∇p

∇ · v = 0 .

We may now use a local representation for v in terms of the Oseen-Fujita

truncated fundamental tensor (Section VII.6):

(x) = −

(x)

(x − y)D

(y)dy

:= −

(x)

(R)

(x − y)D

∗

(y) − w

(y) − u

(y))dy

(XII.7.1 0)

where

(R)

(x − y)| ≤ CR

−3/2

for all suﬃciently large R and with C independent of R. Recalling the summa -

bility properties of w

∗

, w and u and using this latter inequali ty and the H¨older

inequality at the right hand side of (XII. 7.10) for various values of α, we can

easily show the validity of (XII.7.9). For instance, with |α| = 2 we ﬁnd that

(x)| ≤ C

−3/2

2(1 −1/q)

(|v|

+ |u|

) + R

−3/2

2(1 −1/q)

|w|

2,q

and so, noticing that −3/2 + 2(1 − 1/s) < 0 for all s < 4, we prove the

ﬁrst relation in (XII.7.9) by letting R → ∞ in thi s last inequality. The other

relations in (XII.7.9) follow in a similar manner. From (XII.7.9) and (XII.7.8)

we then obtain

+ kw

2q/(2−q)



∂w

∂x



≤ c

(kak

∞

+ kAk

)



∂w

∂x



+ kw

/(2−

)

and so, if

kak

∞

+ kAk

, (XII.7.11)

we conclude

(w − u) ≡

∂(w − u)

∂x

≡ 0, w

≡ u

and the lemma follows from the properties o f w and the fact that u

tends to

zero as |x| tends to inﬁnity.

860 XII Two-Dimensional Flow in Exterior Domains

Remark XI I.7.1 We do not know if the result just shown continues to hold

under the following conditions on u:

∂u

∂x

∈ L

), lim

|x|→∞

u(x) = 0.

Nevertheless, we can treat the case q = 2 if we suppose A ≡ 0, as shown in

the following. 

Lemma XII.7.2 Let u, π be a solution to (XII.7.1) wi th A ≡ 0, such that

u ∈ D

2,2

∂u

∂x

∈ L

Suppose, further,

G ∈ L

), g ∈ W

1,r

), for some r ∈ (1, 2).

Then, there exists a positive constant c = c(r) such that if

kak

∞

< c,

we have

∂u

∂x

∈ L

Proof. Reasoning exactly as in the proof of Lemma XII.7.1, we can show the

existence of a solution w, τ to the problem (XII.7.1) with A ≡ 0, sati sfy ing

w ∈ D

2,r

), τ ∈ D

1,r

∂w

∂x

∈ D

1,r

Letting w = w − u, s = τ − π, it foll ows that

∆w +

∂w

∂x

= a

∂w

∂x

− ∇s

∇· w = 0.

Again as in the proof of Lemma XII.7.1, we may use Lemma II .3.3 to show

that

∂w

∂x

∈ L

)

and so, by Theorem VII.4.1, and by means of the same procedure used in

Lemma XII.7.1, we obtain



∂w

∂x



≤ ckak

∞



∂w

∂x



Therefore, i f kak

∞

is suﬃciently small, we ﬁnd that

w ≡

∂w

∂x

≡ 0,

and the lemma is proved. ut

XII.7 Global Summability of Generalized Solutions when v

∞

6= 0 861

We need another preparatory result.

Lemma XII.7.3 Let v be a D-solution to (XII.0.1) with f of bounded sup-

port and satisfying (XII.3.67) uniformly, with v

6= 0. Then, for suﬃciently

large ρ,

(v − v

) ∈ L

(Ω

) for all s > 16 .

Proof. Without loss of generality, we take v

= −e

and set u := v + e

. We

take R > R

with R

given in Theorem XII.4. 3, and pick x with |x| ≥ 2R.

Taking into account (XII.3.2), we may use the representation formula (V.3. 14)

with u = u

, u

and f = ∂ω/∂x

, −∂ω/∂x

, respectively. After integrating by

parts this formula, w ith the help of (V.3.9) and (V.3.13), we easily deduce the

following inequality

|u(x)| ≤ c

−2+η

(x)

|u(y)|dy +

(x)

|ω(y)|

|x −y|

, (XII.7.12)

where β

(x) := B

(x) − B

R/2

(x), a nd η > 0 is arbitrary. From (XII. 5.1) we

know that there exists R > R

such that |ω(y)| ≤ c

|y|

−3/4

, for all |y| ≥ R.

Since for y ∈ B

(x) it is |y| ≥ |x|−R, and since |x| ≥ 2R, by choosing R > R,

we ﬁnd

|ω(y)| ≤ c

|y|

−3/4

≤ 2c

|x|

−3/4

, y ∈ B

(x) .

Replacing this information back into (XII.7.12) we deduce

|u(x)| ≤ c

−2+η

(x)

|u(y)|dy + |x|

−3/4

= c

−2+η

(x)

|u(y)|dy + |x|

−3/4

(XII.7.1 3)

Furthermore, by Schwarz inequality, Theorem XII.4.3 , and recalling |x| ≥ 2R,

we obtain

(x)

|u(y)|dy ≤

Ω

|u|

|y|

1+ε

1/2



|y|

1+ε



1/2

≤ c

(1+ε)/2+1

≤ c

|x|

(1+ε)/2

R ,

where ε is arbitrary in (0, 1) Substituting this latter i nto (XII.7.13) and choos-

ing η = ε/2 it foll ows that

|u(x)| ≤ c



|x|

1/2+ε

−1

+ |x|

−3/4



, |x| ≥ 2R ,

so that, minimizing with respect to R, we conclude |u(x)| ≤ c| x|

(−1+4ε)/8

for

suﬃciently large |x| and arbitrary ε ∈ (0, 1), which concludes the proof o f the

lemma .

862 XII Two-Dimensional Flow in Exterior Domains

With these lemmas in hand, we are now able to establish the main result

of this section, which is the two-dimensional counterpart of the result given

in Theorem X. 6.4 for the three-dimensional case.

Theorem XII.7.2 Let v, p be a solution to the Navier–Stokes system (XII.0.1)

in Ω of class C

, with

f ∈ L

(Ω) with bounded support, v

∗

∈ W

2−1/q

(∂Ω),

for som e q

> 2, all q ∈ (1, q

], and such that

v ∈ D

1,2

(Ω),

lim

|x|→∞

(v(x) + e

) = 0.

(XII.7.1 4)

Then, we have

+ 1) ∈ L

(Ω) for all t

> 3,

∈ L

(Ω) for all t

> 2,

∂v

∂x

∈ L

(Ω) for all t

> 3/2,

∂v

∂x

, ∇v

∈ L

(Ω) for all t

> 1,

(p − p

) ∈ L

(Ω) for all t

> 2,

where p

is a constant.

Proof. From (XII.7.14)

and from the regularity results near the bounda ry

for the solutions to the Stokes problem (Theorem IV.5.1), we easily deduce

v ∈ W

2,q

(Ω

), p ∈ W

1,q

(Ω

), (XII.7.15)

for a ll r > δ( Ω

). Then, by Lemma XII.3.2 it follows that

∇v ∈ W

1,2

(Ω). (XII.7.16)

From the assumption, (XII.7.15), (XII.7.16 ), and (XII.0. 1) we also have

∇p ∈ L

(Ω). (XII. 7.17)

Now, for R ≥ ρ, let ψ

be a smooth “cut-oﬀ” function deﬁned by

(x) =

(

0 if |x| < R/2

1 if |x| ≥ R.

Setting

u = ψ

(v − e

) ≡ ψ

v, π = ψ

from (XII.0.1) we deduce that u, π satisfy the following system in R

XII.7 Global Summability of Generalized Solutions when v

∞

6= 0 863

∆u +

∂u

∂x

= (ψ

R/2

)

∂u

∂x



R/2

∂v

∂x



+ ∇π + G

∇ · u = g,

(XII.7.1 8)

where

= ψ

f + 2∇ψ

· ∇v + ∆ψ

v −

∂ψ

∂x

v −v

∂ψ

∂x

− p∇ψ

g = v · ∇ψ

Clearly, we have

∈ L

), g ∈ W

1,q

) f or all q ∈ (1 , q

Moreover, we observe that in view of the assumption, by taking R suﬃciently

large, the quantities

kψ

R/2

∞



R/2

∂v

∂x



can b e made less than any prescribed constant. Setting

q =

2 + s

(< 2),

with s given in Lemma XII.7.3, by the H¨older inequality and assumption we

ﬁnd that

∂v

∂x

∈ L

and so by Lemma XII.7.2 with a = ψ

R/2

, G = G

+ u

∂v

∂x

, r = q, and by

(XII.7.1 6), (XII.7.1 8), and by the properties of ψ

, we deduce

∂v

∂x

∈ L

(Ω

From this and (XII.7.15) we ﬁnd

∈ L

2q/(2−q)

), D

∂u

∂x

∈ L

). (XII.7.19 )

We next apply Lemma XII.7. 1 with a = ψ

R/2

, A = ψ

R/2

∂v

∂x

and G = G

In view of (XII.7.14)

, (XII.7.18), (XII.7.1 9), and the properties of ψ

we ﬁnd

that for any gi ven q ∈ (1, 3/2) there exists a suﬃciently large R such that

864 XII Two-Dimensional Flow in Exterior Domains

+ 1) ∈ L

3q/(3−2q)

(Ω

)

∈ L

2q/(2−q)

(Ω

)

∂v

∂x

∈ L

3q/(3−q)

(Ω

)

∂v

∂x

, ∇v

∈ L

(Ω

)

(p − p

) ∈ L

2q/(2−q)

(Ω

)

These conditions along with (XII.7.15), (XII.7.16)–(XII.7.17), and Theorem

II.3.4, allow us to conclude the validity of the summability properties stated

in the theorem. ut

Remark XI I.7.2 By the same method of proo f, we can also show for al l

s ∈ (1, 3/2) that

v ∈ D

2,s

(Ω), p ∈ D

1,s

(Ω).

Therefore, i f, in addition to the assumptio ns of T heorem XII.7.2, we supp ose

Ω of class C

and

∗

∈ W

3−1/r,r

(∂Ω), some r > 2, (XII.7.20)

from Theorem X.1.1 and Theorem XII.3.3 we obtain

v ∈ D

2,τ

(Ω), p ∈ D

1,τ

(Ω), for all τ > 1. (XII.7.21)



We shall next draw some interesting consequences of Theorem XII.7.2.

First of all, we have the following theorem.

Theorem XII.7.3 Let the assumptions of Theorem XII.7.2 be satisﬁed.

Then v and the corresponding pressure ﬁeld p ob ey the energy equati on

(X.2.29).

Proof. By Theorem XII.7.2,

v + v

∞

∈ L

(Ω) ∩ L

(Ω), for all q > 3. (XII.7.22)

In addition,

∂v

∂x

∈ L

(Ω), for all s > 1,

and so, in particular,

∂v

∂x

∈ L

(Ω). (XII. 7.23)

Therefore, the result fol lows from (XII.7.22), (XII.7.23), Theorem X.2.2 and

Exercise X.2.2. ut

XII.7 Global Summability of Generalized Solutions when v

∞

6= 0 865

Theorem XII.7.3 at once gives the following corollary.

Corollary XII.7.1 The solution v, p determined in Theorem XI I .5.1 satisﬁes

the energy equality (X.2.29).

Another consequence of Theorem XII.7.2 is that v and p can be repre-

sented in a way analogo us to that determined for the three-dimensional case.

Speciﬁcally, we have the following.

Theorem XII.7.4 Let the assumption of Theorem XII.7.2 be satisﬁed. As-

sume, further, Ω of class C

and

f ∈ L

(Ω), v

∗

∈ W

2−1/r,r

(∂Ω) t ∈ (1, ∞), r ∈ (2, ∞) .

Then, setting u = v + v

∞

, the following representations hold for all x ∈ Ω

(x) = R

Ω

(x − y)(f

(y) + u

(y)D

(y))dy

∂Ω

(y)T

, e

)(x −y)

−E

(x − y)T

(u, p)(y) − Ru

(y)E

(x − y)δ

dσ

(XII.7.2 4)

and

(x) = R

Ω

(x −y)f

(y)dy − R

Ω

(y)u

(y)D

(x − y)dy

∂Ω

(y)T

, e

)(x − y) − E

(x − y)(T

(u, p)(y)

−Ru

(y)u

(y)) − Ru

(y)E

(x − y)δ

dσ

(XII.7.2 5)

and

p(x) = p

−R

Ω

(x − y)(f

(y) + u

(y)D

(y))dy

∂Ω

(x − y)T

(u, p)(y) −2u

(y)

∂

∂x

(x − y)

−R[e

(x − y)u

(y) − u

(y)e

(x − y)δ

dσ

(XII.7.2 6)

In these relations E, e is the Oseen f unda mental solution, whi le w

is deﬁned

in (VII.6.3) and p

is a constant. All volume integrals in (XII.7 .24)–(XII.7.26)

are absolutely convergent.

Proof. The proof is, in fact, completely analogous to (and, in some steps, even

simpler than) that given in Theorem X.5.2 for the three-dimensional case; it

is l eft to the reader as an exercise. ut