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

Подождите немного. Документ загружается.

906 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

Remark XI II.2.1 Every weak solution to Leray’s problem (corresponding

to the ﬂux Φ) is in the class C

. 

Lemma XIII.2.2 Let Ω be as i n (XIII.1.1 ) and let u, w ∈ D

1,2

(Ω). Then if

v ∈ D

1,2

(Ω) we have



v ·∇u, w



| ≤ c

|v|

1,2

|u|

1,2

|w|

1,2

(XIII.2.7)

with c

= c

(Ω, n). Moreover, if v ∈ D

1,2

(Ω), we have



v · ∇u, w



= −



v · ∇w, u



(XIII.2.8)

so that



v · ∇u, u



= 0. (XIII.2 .9)

Finally, if v ∈ C

, and

≡ |v − v

(i)

1,2,Ω

there is c

= c

(Ω, n) > 0, i = 2, 3 such that



u · ∇v, w



| ≤ c



i=1

+ |v|

1,2,Ω

+ |Φ|



|u|

1,2

|w|

1,2



v ·∇w, u



| ≤ c



i=1

+ |v|

1,2,Ω

+ |Φ|



|u|

1,2

|w|

1,2



v · ∇v, u



| ≤ c



i=1

+ |v|

1,2,Ω

+ |Φ|



|u|

1,2

(XIII.2.10)

If, in addition, ∇ · v = 0, then

(v · ∇u, u) = 0. (XIII.2.11)

Proof. We spl it (v · ∇u, w) as the sum o f three integrals I

, I

, and I

over

the regions Ω

, Ω

, and Ω

, respectively. We have, by the H¨older inequal ity,

| ≤ kvk

4,Ω

kwk

4,Ω

|u|

1,2

. (XIII.2.12)

However, from Lemma XIII.2.1, it follows that

kuk

4,Ω

≤ κ

|u|

1,2

for all u ∈ D

1,2

(Ω), i = 1, 2, (XIII.2.13)

and so (XIII.2.12) yields

| ≤ κ

|v|

1,2

|w|

1,2

|u|

1,2

. (XIII.2.14)

Likewise, we prove

| ≤ κ

|v|

1,2

|w|

1,2

|u|

1,2

. (XIII.2.15)

XIII.2 On the U niqueness of generalized Solutions to Leray’s Problem 907

From the embedding T heorem II.3.4 it follows that

| ≤ kvk

4,Ω

kwk

4,Ω

|u|

1,2

≤ ckvk

1,2,Ω

kwk

1,2,Ω

|u|

1,2

However, since v and w vanish (in the trace sense) at the boundary ∂Ω,

from (II. 5.18) we ﬁnd that k·k

1,2,Ω

and |·|

1,2,Ω

are equivalent norms so that

the latter inequality furnishes, in particular,

| ≤ c|v|

1,2

|w|

1,2

|u|

1,2

. (XIII.2.16)

Relation (XIII.2.7) is a consequence of (XIII.2.14)–(XIII.2.16). To show

(XIII.2.8) we notice that it is trivially veriﬁed (by integration by parts) if

w, v ∈ C

∞

(Ω). Under the assumptions on w, v stated in the theorem, condi-

tion (XIII.2.8) follows from the continuity property (XIII.2.7) and the density

of C

∞

(Ω) into D

1,2

(Ω). Of course, (XIII.2.8) i mplies (XIII.2 .9). Let us now

prove (XIII.2.10). We split, as before, (u · ∇v, w) as the sum of I

, I

, and

. We have

| ≤ |(u · ∇(v − v

(i)

), w)

Ω

| + |(u · ∇v

(i)

, w)

Ω

Thus, by (XIII.2.13) and the H¨older i nequality, we deduce that

| ≤ κ

|u|

1,2

|w|

1,2

∞

(1)

1,2,Σ

kuk

4,Σ

kwk

4,Σ

where integration is performed over the x

variable. Because of Exercise

VI.0. 1, (XIII.2.3), and (XI II.2.4), it readily follows that

| ≤ c(A

+ |Φ|)|u|

1,2

|w|

1,2

. (XII I.2.17)

Likewise, we show

| ≤ c(A

+ |Φ|)|u|

1,2

|w|

1,2

. (XII I.2.18)

Concerning I

, we have

| ≤ c|v|

1,2,Ω

kuk

1,2,Ω

kwk

1,2,Ω

where use has been made o f the H¨older inequality and the embedding Theorem

II.3.4. However, as already remarked, k · k

1,2,Ω

and | · |

1,2,Ω

are equivalent

norms for functions from D

1,2

(Ω

) and so

| ≤ c|v|

1,2,Ω

|u|

1,2

|w|

1,2

, (XIII.2.19)

and (XIII.2.10)

becomes a consequence of (XIII.2.17)–(XIII.2 .19). The pro of

of (XIII.2.10)

is very much the same as that just furnished for (XIII.2.10)

Throughout t he rest of the proof , the symbol c will denote a quantity depending

(at most) on Ω and n.

908 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

and, therefore, it will be omitted. Also the proof of (XIII.2.10)

is obtained

in a similar fashion, once we notice that

(i)

· ∇v

(i)

≡ 0 in Ω

i = 1, 2.

To show the last property (XIII. 2.11), we observe that, since by (XIII.2.10)

v · ∇u · u ∈ L

(Ω),

it follows that

Ω

v · ∇u ·u = lim

R→∞

Ω

0,R

v · ∇u · u (XIII.2.20)

with Ω

0,R

is deﬁned in (XIII.2.1). By integration by parts, for all R > 0,

Ω

0,R

v ·∇u ·u =

(R)

v · nu

(R)

v · nu

(XIII.2.21)

where Σ

(R) [respectively Σ

(R)] denotes the cross section Σ

[respectively

] calculated at x

= −R [respectively x

= R] in the system of coordinates

that Ω

[respectively Ω

] is referred to . Since

≡

(R)

v ·nu

(R)

(v − v

(1)

) · nu

(R)

(1)

· nu

it follows that

| ≤ (kv − v

(1)

2,Σ

(R)

+ kv

(1)

2,Σ

(R)

)kuk

4,Σ

(R)

By Remark XIII.1.1 and Exercise VI.0.1 we have

(kv − v

(1)

2,Σ

(R)

+ kv

(1)

2,Σ

(R)

) ≤ c

for som e c

independent of R. Therefore,

| ≤ kuk

4,Σ

(R)

. (XIII.2.22 )

By the trace inequality of Theorem II.4.1 and (XIII.2.4) we readily see tha t

kuk

4,Σ

(R)

≤ ckuk

1,2,Ω

≤ c

√

µ|u|

1,2,Ω

which, along with (XIII.2.2 2) and the condition u ∈ D

1,2

(Ω), impl ies

lim

R→∞

= 0.

Likewise, one shows

lim

R→∞

(R)

v · nu

= 0

so that, by this relation, (XIII.2.20), and (XIII.2.21) we conclude the validity

of (XIII. 2.11). The lemma is proved. ut

XIII.2 On the U niqueness of generalized Solutions to Leray’s Problem 909

Remark XI II.2.2 Explicit va lues for the constants c

, i = 1, 2, 3 appearing

in the statement of Lemma XIII.2.2 can be of some interest. In particular,

is related to uniqueness conditions for generalized solutions; see Theorem

XIII.2.1. Even though it appears diﬃcult to give an estima te valid for a general

domain Ω, this is actually possible if Ω is of special shape. For instance, if Ω

is an inﬁnite straight cylinder o f circular cross section Σ, one can show

= 2 max



κ,

√

|Σ|

1/2



where κ is the constant g iven in (XIII.2.2) while c

is the Poiseuille constant

deﬁned in Exercise VI.0.1. 

We are now in a position to show the mai n result of this section.

Theorem XIII.2.1 Let v be a generalized solution to Leray’s problem

(XIII.1.2), (XIII.1 .4) corresponding to the ﬂux Φ. If

i=1

|v − v

(i)

1,2,Ω

+ |v|

1,2,Ω

+ |Φ| <

with c

given in (XIII.2.10)

, then v is the only generalized solution corre-

sponding to Φ.

Proof. Let v

be another generalized solution corresponding to Φ. Setting

u = v

−v

from (XIII.1.5 ), it follows that

ν(∇u, ∇ϕ) + (u · ∇u, ϕ) + (u · ∇v, ϕ) + (v · ∇u, ϕ) = 0 (XIII.2.23)

for a ll ϕ ∈ D(Ω). It is readily shown tha t

u ∈ D

1,2

(Ω). (XIII .2. 24)

Actually, i n Ω

, i = 1, 2,

u = (v

− v

(i)

) −(v − v

(i)

)

and, as a consequence o f (i), (iv), and (v) of Deﬁnition XIII.1.1 and in view

of Exercise VI.1.1, it follows that

u ∈ D

1,2

(Ω).

Since u is solenoidal, we have, i n fact

u ∈

1,2

(Ω);

910 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

cf. Section III. 5. However, by Exercise III.5.1,

1,2

(Ω) = D

1,2

(Ω)

and (XIII.2.24) is proved. By virtue of Lemma XIII.2.2 we can now extend

(XIII.2.23) to all ϕ ∈ D(Ω) and, because of (XIII. 2.24), we may take ϕ = u.

By properties (XIII.2.9) a nd (XIII.2 .11),

(v · ∇u, u) = (u · ∇u, u) = 0,

and so (XIII.2.23) with ϕ = u delivers

ν| u|

1,2

+ (u · ∇v, u) = 0. (XIII.2 .25)

Employing (XIII.2.10)

, we ﬁnd

|(u · ∇v, u)| ≤ c

i=1

|v − v

(i)

1,2,Ω

+ |v|

1,2,Ω

+ |Φ|

|u|

1,2

and the theorem fo llows from this i nequality and (XIII.2.25). ut

Remark XI II.2.3 In the case when Ω is an inﬁnite straight cylinder an

estimate from above for the constant c

is given in Remark XIII. 2.1. 

XIII.3 Exi stence and Uniqueness of Solutions to Leray’s

Problem

Existence wi ll be proved by the same Galerkin technique employed in Chapters

IX–XI for analogo us q uestions in domains with compact boundaries. To this

end, we need, as in the linear case, a suitable extension of the Poiseuille

velocity ﬁelds v

(i)

, i = 1, 2. However, since in the case at hand the equations

are nonlinear, we have to face the same problem we already encountered in

Section IX.4 and Section X.4. Actually, let us denote by a an extension of

(i)

, i = 1, 2, that is, a suﬃciently smooth sol enoidal vector ﬁeld in Ω that

vanishes on ∂Ω, equals v

(i)

at l arge distances in Ω

, i = 1, 2, and such that

a · n = Φ.

We then look for a solution to (XIII.1.2)–(XIII.1.4) of the form

v = u + a, u ∈ D

1,2

As we have learned from the Galerkin technique, for such a solution to exist

we need an a priori bound for |u|

1,2

depending only on the data. Replacing

formally this v into (XIII.1.2) we ﬁnd

XIII.3 Existence and Uniqueness of Solutions to Leray’s Problem 911

ν∆u = u · ∇u + u · ∇a + a · ∇u + ∇p − ν∆a + a · ∇a

∇ · u = 0







in Ω

u = 0 at ∂Ω

u ·n = 0

lim

|x|→∞

u = 0 in Ω

Dot-multiplying the ﬁrst equati on by u , which, according to the Galerkin

method, can be a ssumed to be a member of D(Ω), integrating by parts and

employing the boundary conditions we formally derive the following identity

Ω

∇u : ∇u = −

Ω

u · ∇a · u − ν

Ω

∇a : ∇u −

Ω

a · ∇a · u.

Using the results of Lemm a XIII.2.2 along with the properties of a , it is not

diﬃcult to show that the latter identity leads to the following estimate

Ω

∇u : ∇u ≤ −

Ω

u · ∇a · u + C



Ω

∇u : ∇u



1/2

with C depending only on the data. Thus, exactly as in the case of a region

of ﬂow with a compact boundary, if we want to prove existence without re-

strictions from below on the kinematical viscosity ν, we should show tha t for

any α ∈ (0, ν) there exists a(x; α) such that

−

Ω

u · ∇a · u < α

Ω

∇u : ∇u for all u ∈ D(Ω).

However, the existence of such an extension is not known and, perhaps, in

view of wha t we have seen in the case of a bo unded region of ﬂow, it may

not hold. Nevertheless, in place of the preceding property, one does prove the

existence of an extension a verifying



Ω

u · ∇a · u



< c

−1

|Φ|

Ω

∇u : ∇u for all u ∈ D(Ω)

for some c = c(Ω, n), which, therefore, furnishes the desired bound on |u|

1,2

provided

|Φ| < cν.

As a consequence, the question of existence of solutions to Leray’s problem

for arbitrary values of ν or (equivalently) for arbitrary values of the ﬂux Φ

remains open.

We shall now pass to the construction of the ﬁeld a that will satisfy this

property.

See also the proof of Theorem XIII.3.2.

912 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

Lemma XIII.3.3 There exi sts a ﬁeld a : Ω → R

such that

(i) a ∈ W

2,2

loc

(Ω);

(ii) ∇ · a = 0 in Ω;

(iii) a = 0 at ∂Ω;

and, for som e R > 0,

(iv) a = v

(i)

in Ω

;

(v) |a|

1,2,Ω

0,R

≤ c|Φ|;

with Ω

and Ω

0,R

deﬁned in (XIII.2.1) and with c = c(Ω, n, R).

Proof. The ﬁeld a is constructed exactly as in Section VI.1. The only thing

to prove is condition (v). However, we know that in Ω

0,R

a satisﬁes

|a|

1,2,Ω

0,R

≤ c

i=1

(i)

1,2,Σ

with c = c(Ω

0,R

, n), so that (v) follows from this inequality and Exercise

VI.0. 1. ut

The preceding lemma allows us to show the following existence result.

Theorem XIII.3.2 There is a constant c = c(Ω, n) > 0 such that if

|Φ| < cν, (XIII.3.1)

Leray’s problem (XIII.1.2)–(XIII.1.4) admits at least one g eneralized solution

v. Moreover, for some positive C = C(Ω, n),

i=1

|v − v

(i)

1,2,Ω

+ |v|

1,2,Ω

≤ C(1 +

)(|Φ|+ |Φ|

). (XIII.3.2)

Proof. We look for a solution of the form

v = u + a

with a given in Lemma XIII.3.3 and u ∈ D

1,2

(Ω) satisfying

ν(∇u, ∇ϕ) − (u · ∇ϕ, u) = −(u · ∇a, ϕ) + (a · ∇ϕ, u)

−(a · ∇a, ϕ) + ν(∆a, ϕ).

(XIII.3.3)

It is clear that v satisﬁes all the requirements (i)-(v) of Deﬁnition XIII.1. 1.

A solution to (XIII. 3.3) is determined via the Galerkin method. Thus, let

{ϕ

} ⊂ D(Ω)

denote a sequence whose linear hull is dense in D

1,2

(Ω). By Lemma VII.2.1

and the embedding Theorem II.3.4, we can take {ϕ

} satisfying the following

conditions:

XIII.3 Existence and Uniqueness of Solutions to Leray’s Problem 913

(i) (ϕ

, ϕ

) = δ

;

(ii) Given ϕ ∈ D(Ω) and ε > 0, there are a positive integer m = m(ε) and

real numbers γ

, . . ., γ

such that

kϕ −

i=1

ϕk

< ε.

A sequence of “approximating solutions” {u

} to (XIII. 3.3) is then sought of

the type

k=1

ν(∇u

, ∇ϕ

) −(u

· ∇ϕ

, u

) = −(u

·∇a, ϕ

) + (a · ∇ϕ

, u

)

−(a · ∇a, ϕ

) + ν(∆a, ϕ

(XIII.3.4)

with k = 1, . . ., m. Existence to (XIII.3.4), for each m ∈ N, can be established

exactly as in Theorem IX.3.1 and Theorem X.4.1 (see Lemma IX.3.2), pro-

vided we show a suitable bound for |u

1,2

. To obtain this bound, we multiply

(XIII.3.4)

by ξ

and sum over k f rom 1 to m. Recalling that, by Lemma

XIII.2.2, for all m ∈ N,

· ∇u

, u

) = (a · ∇u

, u

) = 0, (XIII.3 .5)

we ﬁnd

ν| u

1,2

= −(u

· ∇a, u

) + (a · ∇a, u

) + ν(∆a, u

). (XIII.3.6)

From Lemma XIII.2.2 and Lemma XIII.3.3 it foll ows that

|(u

·∇a, u

)| ≤ c

|Φ| |u

1,2

|(a · ∇a, u

)| ≤ c

|Φ|

1,2

(XIII.3.7)

with c

= c

(Ω, n). Furthermore, from (VI.1.9) and (VI.1.10 ) we have

|(∆a, u

)| = |(∇a, ∇u

)

Ω

0,R

and so, again by Lemma XIII.3.3

|(∆a, u

)| ≤ c

|Φ| |u

1,2

, (XI II.3.8)

with c

= c

(Ω, n). Thus, if we take, for instance,

|Φ| <

ν,

using (XIII.3.5)–(XIII.3.8) a nd Lemm a IX.3.2 we show existence to problem

(XIII.3.4) for all m ∈ N. Furthermore,

914 XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

1,2

≤

(|Φ|+ |Φ|

). (XIII.3 .9)

Using (XIII.3.9) and the weak compactness property of the spaces

1,2

(Ω),

we may ﬁnd a subsequence {u

} and a vector ﬁeld u ∈ D

1,2

(Ω) such that

→ u weakly in D

1,2

(Ω)

→ u strongly in L

(Ω

), for all bo unded Ω

⊂ Ω.

Reasoning as in Theorem IX.3.1 and Theorem X. 4.1 and taking advantage

of condition (ii) on {ϕ

}, we then show that u satisﬁes (XIII.3 .3) for al l

ϕ ∈ D(Ω). Since in view of (XIII.3.9) and the property of weak limits, we

have

|u|

1,2

≤

(|Φ| + |Φ|

with the help of Lemma XIII.3. 3 and Exercise VI.0.1 we also have

|v − v

(i)

1,2,Ω

≤ |a − v

(i)

1,2,Ω

+ |u|

1,2

≤ |a|

1,2,Ω

0,R

+ |v

(i)

1,2,Ω

∩Ω

0,R

(|Φ|+ |Φ|

)

≤ c



1 +



(|Φ| + |Φ|

(XIII.3.10)

Likewise, again by Lemma XIII.3.3 and Exercise VI.0.1, we ﬁnd

|v|

1,2,Ω

≤ |a|

1,2,Ω

+ |u|

1,2

≤ c



1 +



(|Φ|+ |Φ|

Therefore, estimate (XIII.3.2) follows from this inequality and (XIII.3.10),

and the theorem is proved. ut

Remark XI II.3.4 Theorem XIII.3.2 states, in particular, that, unlike the

linearized case, the nonlinear Leray’s problem is solvable under the restriction

(XIII.3.1) for the ﬂux Φ. The investigation of whether this restriction can be

removed is, undoubtedly, one of the most challenging problems in theoretical

ﬂuid dynamics. In this respect, it should be observed that if we relax require-

ment (v) of Deﬁnition XIII.1.1, that is, if we do not im pose a priori that

the solution converges to the corresponding Poiseuille ﬂows in the outl ets Ω

then one can show existence of solutions for arbitrary values of Φ. Speciﬁcally,

Ladyzhenskaya & Solonnikov (1980, Theorems 3.1, 3.2) have shown

that for

any Φ ∈ R there exists a pair v , p obeying (XIII.1.2)–(XIII.1.4). Co ncerning

the behavior at inﬁnity, this solution satisﬁes the following conditions:

Actually, in a class of domains larger (for shape) than that considered by us here.

See ( XIII.2.1) for the deﬁnition of the domains involved in (XIII.3.11).

XIII.3 Existence and Uniqueness of Solutions to Leray’s Problem 915

Ω

∇v : ∇v ≤ c

t, for all t > 0

Ω

t,t+1

∇v : ∇v ≤ c

for a ll t > 0,

(XIII.3.11)

with c

and c

independent of t. Moreover, v , p is unique if |Φ| is “ suﬃciently

small.” These results resemble in a sense those for D-solutions of the two-

dimensional nonlinear exterior problem, since in the case at hand the main

problem also remains the investigation of the asymptotic b ehavior, starting

with a certain regularity at la rge distances, here expressed by (XIII. 3.11). In

particular, denoting by S

= S

(Φ, Σ

), i = 1, 2, the “lim it sets,” i.e., the set

constituted by those vector ﬁelds that the solution v satisfying (XIII.3.11)

tends to eventually as |x| → ∞ in Ω

, one should investigate if S

= {v

(i)

}. 

Remark XI II.3.5 Also in view of what was observed in the preceding re-

mark, it appears of a certain interest to determine an explicit and p ossibly

sharp value for the constant c entering condition (XIII.3.1). In this respect

we have a result due to Amick (1977) that ensures that, if Ω

is simply con-

nected, c depends only on Ω

(through their sections Σ

) and not on Ω

; see

Amick, loc. cit. Theorem 3 .6. Moreover, c can be determined in an “optimal”

way by solving a suitable variational problem strictly related to the nonlinear

stabili ty property of Poiseuille ﬂow and, if the cross sections Σ

are of special

shape, c can be explicitly evaluated. For instance, if Σ

is a circle of radius

, we have

c = 127.9 m in{R

, R

};

see Amick loc. cit. §3.4. 

Exercise XIII.3.1 (Generalizati on of Theorem XIII.3.2 to domains with more

tha n two cylindrical ends). Assume that instead of two exits to inﬁnity, Ω

and Ω

the domain Ω has m ≥ 3 exits Ω

, . . . , Ω

, where Ω

, . . . , Ω

can be represented as

Ω

(“upstream” exits) and Ω

j+1

, . . . , Ω

as Ω

(“downstream” exits). Assume also

that

Ω −

[

i=1

Ω

is bounded and that Ω is of class C

∞

. Denote by Φ

the ﬂuxes in Ω

. Then show

that for every choice of Φ

satisfying the compatibility condition of zero total ﬂux,

i.e.,

i=1

i=j+1

there is a c = c( Ω, n) > 0 such that if

i=1

|Φ

| < cν

Leray’s problem is solvable in Ω.