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

Подождите немного. Документ загружается.

666 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

(k,ρ)

1/2

≤ c



kuk

2,B

)

+ k∇uk

2,B

)



≤ c



kuk

4,B

)

+ k∇uk

2,B

)



with c

= c

(n, k), and so (X.2.23) with i = 1 is recovered. By an analog ous

argument one shows (X. 2.23) with i = 2. Putting

= {x ∈ Ω : |x

| < k, x

∈ R},

from (X.2.22), and (X.2.23) we ﬁnd

∞

· ∇u · u = 0 (X.2.24)

for a ll suﬃciently large k. On the other hand, by (X.2.21),

lim

k→∞

∞

·∇u ·u =

Ω

∞

· ∇u · u

and so, by (X.2.24),

∞

·∇u, u) = 0,

which completes the proof of the theorem. ut

Remark X.2.2 The theorem continues to hold under diﬀerent assumptions

on f . In fact, f is required to satisfy

[f , v

−a] → [f, v − a].

Therefore, one can take

f ∈ L

4/3

(Ω), (X.2.25)

or else, if v

∞

6= 0,

f ∈ L

(Ω).



Remark X.2.3 If n = 3, Ω is of class C

, and f satisﬁes (X.2.25), then

condition (X.2.10) alone is enough to ensure the validity of the energy equality

also in the case v

∞

6= 0. In fact, under the stated assumptions on v a nd f,

by the H¨older inequality we ﬁnd [(v + v

∞

) · ∇v + f ] ∈ L

4/3

(Ω). Therefore,

from Theorem VII.7.1 and Theorem VII. 6.2, it easily follows, in particular,

that v

∞

·∇v ∈ L

4/3

(Ω), and condition (X.2.11) remains satisﬁed with q = 4.



Remark X.2.4 Theorem X.2.1 continues to hold, as it stands, in dimension

n ≥ 4 , the proof remaining completely unchanged. In this respect, it is in-

teresting to observe that if n = 4, from Theorem II.6.1 it follows that every

generalized solution corresponding to v

∗

≡ v

∞

≡ 0 satisﬁes (X.2.10). We may

then conclude that in dimension 4 any generalized solution corresponding to

∗

≡ v

∞

≡ 0 satisﬁes the energy equality. Such a result is not known in

dimension n = 3. 

X.2 On the Validity of the Energy Equation for Generalized Solutions 667

Theorem X.2.1 can be extended, with no essential technical changes, to the

case when the velocity v

∗

at the boundary is not identically zero. To this end,

we need a suitable extension of both v

∗

and v

∞

. Following what we already

did in the proof of Theorem V.2.1, for Ω locally Lipschitz and

∗

∈ W

1/2,2

(∂Ω),

we may take

A = V + Φ∇E − a. (X.2.26)

Here V ∈ W

1,2

(Ω) is an extension of v

∗

− Φ∇E vanishing in Ω

for some

ρ > δ(Ω

), Φ is the ﬂux of v

∗

through ∂Ω, and a is an extension of v

∞

in the sense of (X.2.7 ). Taking into account the elementary properties of the

fundamental solution of Laplace’s equation, from (X.2.2 6) it i s apparent that

(i) |A(x) + v

∞

| ≤ c|Φ||x|

−n+1

in Ω

, c = c(n);

(ii) A ∈ D

1,s

(Ω

) ∩ W

1,2

(Ω

) ∩ C

∞

(Ω

), for all s > 1;

(iii) A = v

∗

at ∂Ω;

(iv) ∇ · A = 0 in Ω.

(X.2.27)

In particular, if Φ = 0, A + v

∞

is of bounded support in Ω. Any ﬁeld A

satisfying conditions (i)-(iv) in (X.2.27) will be called an extension of v

∗

and

∞

. Thus, generalizing Deﬁnition X.2.1, we gi ve the following deﬁniti on.

Deﬁnition X.2.2. Let v be a generalized solution to (X.0.8), (X.0.4) and let

A be an extension of v

∗

and v

∞

. The relati on

|v|

1,2

+ R[f, v − A] = (∇v, ∇A) − R(v · ∇A, v − A) (X.2. 28)

is called the generalized energy equation.

As expected, under suitable regula rity assumptions o n the data and v,

identity (X.2.28) reduces to the energy equatio n in its cl assical formulation,

cf. Exercise X.2.2.

By a procedure completely a nalog ous to tha t used in the proof of Theorem

X.2. 1, one can show the following result.

Theorem X.2.2 Let v be a generalized solution to the Navier–Stokes prob-

lem (X.0.8), (X.0.4) in an exterior locally Lipschitz domain of R

, n = 2, 3,

corresponding to

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2,2

(∂Ω), v

∞

∈ R

Then, if v

∞

= 0, a suﬃcient conditi on in order that v satisﬁes the generalized

energy equation (X.2.28) for any extension A of v

∗

and v

∞

is that (X.2.10)

holds. L ikewise, if v

∞

6= 0, a suﬃcient condition in order that v satisﬁes

(X.2.28) for any extension A of v

∗

and v

∞

is that (X.2.11) holds for some

q > n/(n −1).

668 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

Remark X.2.5 The restriction on q (not needed in Theorem X.2.1) is due

to the fact that, if Φ 6= 0, the ﬁeld A belongs to L

(Ω

) for q > n/(n − 1)

only. However, if Φ = 0, A can be taken of bounded support and we may

assume q ∈ (1, ∞). 

Remark X.2.6 Remark X.2.2 equally applies to Theorem X. 2.2. The same

holds for Remark X.2.3, provided we a ssume v

∗

∈ W

5/4,4/3

(∂Ω). 

Remark X.2.7 Theorem X.2.2 continues to hol d in dimension n ≥ 4 . 

Exercise X.2.2 Let f , v, and Ω satisfy the assumptions of Exercise X.2.1 and

Theorem X.2.2. Suppose, further, that

∗

∈ W

2−1/r,r

(∂Ω),

with r as in Exercise X. 2.1. Then, if Φ = 0, show the identity

∂Ω

n · T (v, p) · (v

∗

+ v

∞

) −

∂Ω

∗

+ v

∞

)

∗

· n

= R[f , A + v

∞

] + 2(D(v), D(A)) − R(v · ∇A, v − A)

with A an extension of v

∗

and v

∞

. If Φ 6= 0, show the validity of the same identity

under the additional assumption f ∈ D

−1,3

(Ω). Thus, since

|v|

1,2

− (∇v, ∇A) = 2[kD(v)k

−(D(v), D(A))]

under the stated assumptions, in view of Theorem X. 2.2, v ob eys the energy equation

Ω

D(v) : D(v) −

∂Ω

[(v

∞

∗

) ·T −

∗

+ v

∞

)

∗

] ·n + R

Ω

f ·(v + v

∞

) = 0.

(X.2.29)

X.3 Some Uni queness Results

The aim of this section is to determine conditions under which a weak solu-

tion v is unique. The problem oﬀers more or less the same type of technical

diﬃculties encountered in the preceding section, as we are about to explain.

Let us denote by C

the class of weak soluti ons achieving the same data as

v. Then, as we shall prove, in order for a weak solution to be unique in C

in addition to the (expected) restriction on the “size” of v in suitable norms

of the typ e made in Theorem IX.2.1 for Ω bounded, we have to require on

v some extra conditions at large spatial distances that a priori do not follow

directly from Deﬁnition X.1.1. Thus, it becomes necessary to ascertain if such

conditions are met by prescribing the data appropriately. This will be done

in Section X.7 and Section X.9. It should be also said that, in fact, we are

The proof of this identity is similar to that given in footnote 4 in this section.

X.3 Some Uniqueness Results 669

able to prove uniqueness only in the a priori small er class C

constituted by

those elements o f C

that satisfy the energy inequality (cf. (X.3.1)) and, if

∞

6= 0, verify further summability conditions at large distances (cf. (X.3.2)).

Nevertheless, in Section X.4 we will prove that C

is certainly nonempty.

To accomplish our objective, however, we need to employ a method that

is a bit diﬀerent from that adopted for ﬂows in bounded domains. This is

because the use of such a method would lead to a uniqueness result that does

not impose extra conditions directly on v but, rather, on v − w, w ∈ C

, in

contrast to what stated previously. To see why thi s happens, we recall that

the starting point of the method is the identity (IX.2.5) which, according to

the nondimensionalization used in the present chapter, now reads

−1

(∇u, ∇ϕ) + (u · ∇u, ϕ) + (u · ∇v, ϕ) + (v · ∇u, ϕ) = 0.

The next step is to substitute u for ϕ into this relation and this can b e done

via the usual approximating procedure that employs the continuity of the

trilinear form (X.2.6). According to Lemma X.2 .1, we must then require some

extra conditions on u. However, u is the diﬀerence of two generalized solutions

and the method would lead to a uniqueness result diﬀerent from that stated

at the beginning of the current section.

The method we shall adopt here is due to Galdi (1992a, 1992c) and relies

upon an idea introduced by Leray (1934, §32) in a completely diﬀerent context,

namely, that of local regularity of weak solutions to the initial value problem

for the Navier–Stokes equations, and successively generalized by Serrin and

Sather; cf. Serrin (1963, Theorem 6), Sather (1963, Theorem 5.1). In the case

of steady ﬂows in exterior domains with v

∗

≡ v

∞

≡ 0, the method was ﬁrst

considered by Kozono & Sohr (1993).

Before proving the main results, we need to deﬁne the class C

properly.

Deﬁnition X.3.1 C

denotes the subclass of C

constituted by those gener-

alized solutions w satisfying the generalized energy inequality

|w|

1,2

+ R[f , w − A] ≤ (∇w, ∇A) − R(w ·∇A, w − A) (X.3.1)

with A an extension of v

∗

and v

∞

in the sense of Deﬁnition X.2.2. Moreover,

if v

∞

6= 0, we denote by C

v,q

the subclass of C

of functions w such that

w + v

∞

∈ L

(Ω)

∞

· ∇w ∈ L

(Ω)

(X.3.2)

for som e q > n/(n − 1), q

= q/(q − 1).

Remark X.3.1 The condition q > n/(n −1) is required only because, if the

ﬂux Φ of v

∗

through ∂Ω is nonzero, A+v

∞

is not summable at large distances

for q ≤ n/(n − 1) so that the term

(w · ∇A, w − A)

670 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

can be meaningless. On the other hand, if Φ = 0 we can ta ke A − v

∞

bounded support in Ω (see Deﬁnition X.2.2) so that the restriction q > n/(n−

1) can be dropped. 

We are now in a position to prove the following.

Theorem X.3.1 Let Ω be a locally Lipschitz exterior dom ain of R

. Assume

v is a generalized solution to the Navier–Stokes problem (X.0.8), (X.0.4) cor-

responding to data

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2,2

(∂Ω), v

∞

∈ R

and such that

v + v

∞

∈ L

(Ω). (X.3.3)

If v

∞

6= 0, suppose, further, that

v + v

∞

∈ L

(Ω), v

∞

· ∇v ∈ L

(Ω) (X.3.4)

for som e q > 3/2, q

= q/(q − 1). Then, if

kv + v

∞

√

, (X.3.5)

v is the unique soluti on in the class C

if v

∞

= 0 and in the class C

v,q

∞

6= 0.

Proof. Let w be any element from C

or C

v,q

, according to whether v

∞

is zero

or not. The ﬁeld w −A has zero trace a t the boundary, is divergence-free and

vanishes at inﬁnity in the sense of Deﬁnition X.1.1 (iv). Therefore, from the

results of Theorem III.5.1, it follows that

w − A ∈ D

1,2

(Ω). (X.3.6)

Furthermore, if v

∞

6= 0 , from (X.3.2) and (X.2.27)

w − A ∈ L

(Ω). (X.3.7)

By (X.3.6), (X.3.7), and Theorem III.6.2 we may ﬁnd a sequence {ϕ

} ⊂ D(Ω)

converging to w −A in D

1,2

(Ω) ∩L

(Ω). Replacing ϕ

for ϕ into (X.1.2) and

reasoning as in the proo f of Theorem X.2.1 (with sl ight diﬀerence in details)

we deduce tha t

(∇v, ∇w) + R[f , w − A] −(∇v, ∇A) + R(v · ∇v, w −A) = 0. (X.3.8)

Likewise, observing that v −A has the same summability properties of w −A

and that, in addition, by (X.3.3)

v − A ∈ L

(Ω),

X.3 Some Uniqueness Results 671

one shows with no diﬃculty that

(∇w, ∇v) + R[f , v −A] − (∇w, ∇A) + R(w · ∇w, v −A) = 0. (X.3.9)

Finally, noticing that, by Theorem II.6.1,

v + v

∞

∈ L

(Ω),

from (X.3.3) we have by interpolation (cf. (II.2.10))

v + v

∞

∈ L

(Ω),

and by Theorem X.2.2 we conclude

|v|

1,2

+ R[f, v −A] = (∇v, ∇A) − R(v ·∇A, v −A). (X.3. 10)

Addition of (X.3.8) and (X.3.9) yields

−2(∇w, ∇v) −R(v · ∇v, w − A) −R(w · ∇w, v − A) + (∇v, ∇A)

+(∇w, ∇A) − R[f, v − A] −R[f, w −A] = 0.

(X.3.11)

Summing side by side (X.3.1), (X.3.10), and (X.3.11) we deduce

|u|

1,2

≤ (w ·∇w, v−A)+(v·∇v, w−A)−(w·∇A, w−A)−(v ·∇A, v −A)

with u = w−v. Adding and subtracting to the right-hand side of this relation

the quantity

(v ·∇w, v − A) + (u ·∇v, v −A)

(notice that each term in the sum is well-deﬁned), we obtain

|u|

1,2

≤ (u ·∇u, v + v

∞

) − (u · ∇u, v

∞

+ A)

+(u · ∇v, v − A) + (v · ∇w, v − A) − (v · ∇A, v −A)

−(w · ∇A, w − A) + (v · ∇v, w −A).

Since

−(u ·∇u, v

∞

+ A) + (u ·∇v, v −A) = (u ·∇v, v + v

∞

) −(u ·∇w, v

∞

+ A),

it follows that

|u|

1,2

≤ (u · ∇u, v + v

∞

) + (u · ∇v, v + v

∞

)

−(u · ∇w, v

∞

+ A) + (v · ∇(w − A), v −A)

−(w ·∇A, w − A) + (v · ∇v, w − A).

(X.3.12)

The following identities hold

672 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

(i) (u · ∇v, v + v

∞

) = 0;

(ii) (v · ∇v, w − A) = −(v · ∇(w − A), v + v

∞

);

(iii) (w · ∇A, w −A) = −(w · ∇(w − A), A + v

∞

);

(iv) (u · ∇w, v

∞

+ A) = (u · ∇(w − A), v

∞

+ A).

To show (i), we let {u

} ⊂ D(Ω) be a sequence approximating u in D

1,2

(Ω).

Set

Ω

= supp (u

)

and denote by {v

} ⊂ C

∞

(Ω

) a sequence converging, for each ﬁxed k, to

v − v

∞

in W

1,2

(Ω

). Clearly, for all m ∈ N we have

· ∇v

, v

) =

, ∇v

) = 0

and so, passing to the limit m → ∞, by Lemma X.2.1,

· ∇v, v + v

∞

) = (u

·∇(v + v

∞

), v + v

∞

) = 0, for all k ∈ N.

This relation, along with Lemma X.2.1, implies (i). In a similar way, one

proves (ii) and (iii). Finally, consider the identity

(u · ∇w, v

∞

+ A) = (u · ∇(w − A), v

∞

+ A) + (u · ∇(A + v

∞

), A + v

∞

By a reasoning completely analogous to that used before, one shows

(u ·∇(A + v

∞

), A + v

∞

) = 0,

so that also (iv) follows. Replacing (i)-(iv) in (X.3.12) we obtain

|u|

1,2

≤ (u · ∇u, v + v

∞

) − (u ·∇(w − A), v

∞

+ A)

−(v · ∇(w − A), v

∞

+ A) + (w ·∇(w − A), v

∞

+ A)

= (u · ∇u, v + v

∞

(X.3.13)

From Lemma X.2.1 it follows that

|(u · ∇u, v + v

∞

)| ≤

√

|u|

1,2

kv + v

∞

which, once substituted into (X.3.13), furnishes

|u|

1,2

−1

−2(3)

−1/2

kv + v

∞

) ≤ 0. (X.3.14)

Thus, if (X.3.5) holds, (X.3.14) implies u(x) = 0 for a.a. x ∈ Ω, and the

theorem is proved. ut

Remark X.3.2 Theorem X.2.1 i s easily extended to any space dimension

n ≥ 4. Actually, it is enough to replace (X.3.3) with

X.3 Some Uniqueness Results 673

v + v

∞

∈ L

(Ω), (X.3.15)

(X.3.5) with

kv + v

∞

(n − 2)

√

R(n − 1)

, (X.3. 16)

and the condition q > 3/2 in (X.3.4) by q > n/(n − 1). It is interesting to

observe that if n = 4, by Theorem II.6.1, every generalized solution satisﬁes

(X.3.15) and so, if v

∞

= 0, every generalized solution v satisfying (X.3.16) is

unique in the class C

. For the case of plane motions, we refer the reader to

Section XII.2. 

Sometimes, it is convenient to formulate uniqueness theorems with sum-

mabi lity assumptions replaced by pointwise bounds. Actually, such a type

of result is of great relevance when investigating the asymptotic structure of

generalized solutions when v

∞

= 0. Our goa l is then to show a uniqueness

result i n such a direction. For the applications we have in mind, it will be

enough to prove this when v

∗

≡ v

∞

≡ 0. Nevertheless, the result admits a

straightforward extension to the more general case of nonzero v

∗

and v

∞

which we leave to the reader as an interesting exercise.

Theorem X.3.2 Let Ω be a s in Theorem X.3.1 and let v be a generalized

solution to the Navier–Stokes problem (X.0.8), (X.0.4) corresponding to the

data

f ∈ D

−1,2

(Ω) , v

∗

≡ v

∞

≡ 0,

and such that

|x||v(x)| ≤ M, (X.3.1 7)

for a .a. x ∈ Ω and some M > 0. Then, if

M < (2R)

−1

(X.3.18)

v is the unique sol ution in the class C

Proof. Let w be any element from C

. By Deﬁnition X.1.1 and the regularity

of Ω,

w ∈ D

1,2

(Ω).

It is easy to show the validity of (X.3.8) with A ≡ 0. In fact, let {ϕ

} ⊂ D(Ω)

be a sequence converging to w i n D

1,2

(Ω) and set ϕ = ϕ

into (X.1.2).

Evidently, as k → ∞,

(∇v, ∇ϕ

) → (∇v, ∇w)

[f, ϕ

] → [f, w] .

(X.3.19)

Furthermore, from (X.3.17) and Theorem II.6.1 we recover

|(v · ∇v, ϕ

− w)| ≤ M |v|

1,2

k(ϕ

− w)/|x|k

≤ c|v|

1,2

|ϕ

− w|

1,2

674 X Three-Dimensional Flow in Exterior Domains. Irrotational Case

and so

(v · ∇v, ϕ

) → (v ·∇v, w).

Thus, from this latter relation, (X.1.2) with ϕ ≡ ϕ

, and (X.3.19), we then

arrive a t the identity

(∇v, ∇w) + R[f, w] + R(v · ∇v, w) = 0. (X.3.20)

We shall next show the following relation:

(∇w, ∇v) + R[f, v] + R(w · ∇w, v) = 0 . (X.3.21)

To this end we notice that, by Lemma X.1.1, for all ψ ∈ C

∞

(Ω),

(∇w, ∇ψ) + R[f , ψ] + R(w · ∇w, ψ) = (p, ∇ · ψ) (X.3.22)

where p ∈ L

(Ω

), any R > δ(Ω

), is the pressure ﬁeld associated to w. By

this and Lemma IX.1.1, (X.3.22) continues to hold for every ψ ∈ W

1,2

(Ω

Let ψ

(x) be a smooth “cut-oﬀ” function such that ψ

(x) = 1 i f |x| ≤ R,

(x) = 0 if |x| ≥ 2R and, f urthermore,

|∇ψ

(x)| ≤ CR

−1

for som e po sitive C independent of R and x. Taking

ψ = ψ

we ﬁnd

(ψ

∇w, ∇v) + (∇w, v ⊗∇ψ

) + R(w ·∇w, ψ

v) + R[f, ψ

v] = (p, ∇ψ

·v)

(X.3.23)

where we have used the condition ∇ · v = 0 in Ω. Recalling the properties of

we have

|(∇w, v ⊗∇ψ

)| ≤ c |w|

1,2,Ω

R,2R

kv/| x|k

2,Ω

R,2R

|[f, (ψ

− 1)v] | ≤ |f|

−1,2

|(ψ

− 1)v|

1,2

≤ c |f|

−1,2



k(ψ

− 1)∇vk

+ kv/|x|k

2,Ω

R,2R



|((w · ∇w, (ψ

−1)v)| ≤ M k(ψ

− 1)∇wk

kw/|x|k

Employing these inequalities along with Theorem II.6.1 into (X.3.23) and then

letting R → ∞ furnishes

(∇w, ∇v) + R[f, v] + R(w · ∇w, v) = lim

R→∞

(p, ∇ψ

· v). (X.3.24)

To show (X.3.21), it remains to show that the limit on the right-hand side

of (X.3. 24) is zero. To this end, we recall that, from Lemma X.1. 1 and the

assumption on f, we have

X.3 Some Uniqueness Results 675

p = p

+ p

, p

∈ L

(Ω

) , p

∈ L

(Ω

) , ρ > δ(Ω

) . (X.3.25)

Thus, from the H¨older inequality a nd (II.6.48), (II.6.4 9) we have

(R) ≡ | (p, ∇ψ

· v) ≤ kp

3,Ω

R,2R

k∇ψ

·vk

3/2,Ω

R,2R

+kp

2,Ω

R,2R

k∇ψ

·vk

2,Ω

R,2R

≤ c





3,Ω

R,2R

−1

2/3

+ kp

2,Ω

R,2R





|v|

1,2

However, from (X.3.25) it foll ows that

3,Ω

R,2R

+ kp

2,Ω

R,2R

→ 0 as R → ∞

and so

lim

R→∞

(R) = 0.

The validity of (X.3.21) is thus established. We next observe that by (X.3.17)

(and (i) of D eﬁnition X.1.1, Lemma II. 6.1, and Theorem II.3.4 if Ω = R

) it

follows that

v ∈ L

(Ω) (X.3.26)

so that, in view of Theorem X.2.1, v obeys the energy equality

|v|

1,2

= −R[f, v] . (X.3.27)

On the other hand, by assumption, w obeys the energy inequality

|w|

1,2

≤ −R[f, w] . (X.3.28)

Adding the four displayed equations (X.3.20), (X.3.21), (X.3.27), and (X.3.28)

yields

−1

|u|

1,2

≤ (v · ∇ v, w) + (w ·∇w, v) (X.3.29)

with u = w −v. The foll owing identities hold:

(i) (v ·∇v, w) = −(v · ∇w, v);

(ii) (B · ∇v, v) = 0, B = v, w.

To show (i), let {w

} ⊂ D(Ω) be a sequence converging to w in D

1,2

(Ω).

For ﬁxed k ∈ N, we choose ρ > δ(Ω

) with Ω

⊃ supp (w

) and denote by

} ⊂ C

∞

(Ω

) a sequence converging to v in W

1,2

(Ω

). Clearly,

· ∇v

, w

) = −(v

· ∇w

, v

) + (∇ · v

, w

·v

Letting m → ∞ into this identity, with the aid o f Lemma IX.1.1 and the fact

that, as m → ∞,

k∇· v

2,Ω

→ 0,