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

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616 IX Steady Navier–Stokes Flow in Bounded Domains

|a(u, V , u)| ≤

(

η +

m+1

i=1



kσ

n/2,Ω

i,d

+ c

kσ

n,Ω

i,d



|Φ

)

|u|

1,2

with c

and c

suitable constants. Likewise, inequality (IX.4.26) is replaced

by the following one

kV k

1,q

≤ C kv

∗

1−1/q,q(∂Ω)

with C = C(n, Ω, η, M ), provided kv

∗

1−1/q,q(∂Ω)

≤ M . 

Exercise IX.4.2 (Alekseev & Tereshko, 1998) By adopting (and simplifying) the

argument s used in the proof of Lemma IX.4.2, show the following result. Suppose

Ω and v

∗

satisfy the assumption of Lemma IX.4.2 with Φ

= 0, i = 1, . . . , m + 1.

Then, given η > 0 there exists a solenoidal extension, V

∈ W

1,2

(Ω) of v

∗

, such

that

u · ∇V

· u

≤ ηkv

∗

1/2,2(∂Ω)

|u|

1,2

for all u ∈ H

(Ω). Moreover, there is a constant c = c(η, n, Ω) such that

1,2

≤ c kv

∗

1/2,2(∂Ω)

We are now in a position to prove the main results of this section.

Theorem IX.4.1 Let Ω be a bounded locally Lipschitz domain of R

, n =

2, 3, with ∂Ω constituted by m+1 connected compo nents Γ

. . .Γ

m+1

, m ≥ 0,

and let

∗

∈ W

1/2,2

(∂Ω), f ∈ D

−1,2

(Ω),

with v

∗

satisfying (IX.4.24). The following properties hold.

(i) Existence. If

Φ ≡

m+1

i=1



4cκ

kσ

n/2,Ω

i,d

+ κkσ

n,Ω

i,d





∗

·n



< ν ,

(IX.4.50)

there is at least one generalized solution v to problem (IX.0.1), (IX.0.2),

with corresponding pressure ﬁeld p ∈ L

(Ω), associated to v by Lemma

IX.1. 2, that satisﬁes the inequality

kpk

≤ c



|f |

−1,2

+ kvk

1,2

+ νkvk

1,2



(IX.4.51)

where c = c(n, Ω).

(ii) Estimate by the da ta. (a) Let

1/2,2

(∂Ω) := {φ ∈ W

1/2,2

(∂Ω) : kφk

1/2,2(∂Ω)

≤ M }, some M > 0.

(IX.4.52)

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 617

If v

∗

∈ W

1/2,2

(∂Ω) and Φ ≤ ν/2, any generalized solution v correspond-

ing to v

∗

and f satisﬁes the estimate:

kvk

1,2

≤

|f |

−1,2

+ c



∗

1/2,2(∂Ω)

+ kv

∗

1/2,2(∂Ω)



, (IX.4.53)

where c

= c

(n, Ω), while c

= c

(n, Ω, ν, M ) .

(b) There exists c

= c

(n, Ω) such that if

∗

1/2,2(∂Ω)

≤ c

ν/2,

any generalized solution v corresponding to v

∗

and f veriﬁes the follow-

ing estimate

kvk

1,2

≤



|f|

−1,2

+ kv

∗

1/2,2(∂Ω)

+ ν kv

∗

1/2,2(∂Ω)



, (IX. 4.54)

with c

= c

(n, Ω). Estimates for p follow from (IX.4.51)–(IX.4.54).

Proof. We look for a solution of the form v = u+V , where V is the extension

of the ﬁeld v

∗

constructed in Lemma IX.4.2. We then consider a sequence {u

}

of “approximating solutions” as in Theorem IX.3.1 , i.e.,

k=1

ν(∇u

, ∇ψ

) + (u

· ∇u

, ψ

) + (u

· ∇V , ψ

) + (V · ∇u

, ψ

)

= −hf , ψ

i− ν(∇V , ∇ψ

) − (V · ∇V , ψ

), k = 1, 2, . . ., s.

(IX.4.55)

From this po int on we can repeat step by step the proof of Theorem IX.3.1,

provided we show a uniform bound on |u

1,2

. But, as already seen, this is

easily achieved thanks to the particular choice of the ﬁeld V . Actually, multi-

plying (IX.4.55)

by ξ

, summing over k from one to s and recalling (IX.3.8)

we have

ν| u

1,2

+ (u

·∇V , u

) = −hf , u

i−ν(∇V , ∇u

) −(V ·∇V , u

). (IX.4.56)

Using (IX.4.25) with η = (ν − Φ)/2 (say) and Lemma IX.1.1, from (IX. 4.56)

it follows that

(ν −Φ)|u

1,2

≤ (|f|

−1,2

+ C(V , ν)) | u

1,2

which, in view of (IX.4.50), furnishes the desired bound on |u

1,2

. Moreover,

(IX.4.51) is established exactly as in Theorem IX.3. 1. The proof of the exis-

tence can be then considered complete. We shall now show the second part

of the theorem. Let v be a generalized solution corresponding to v

∗

and f .

We write v = w + U, with U ∈ W

1,2

(Ω) solenoidal extension of v

∗

to be

speciﬁed later on. From (IX.1.2) we ﬁnd

618 IX Steady Navier–Stokes Flow in Bounded Domains

ν(∇w, ∇ϕ) + (w · ∇w, ϕ) + (w · ∇U, ϕ) + (U · ∇w, ϕ)

= −hf , ϕi− ν(∇U, ∇ϕ) − (U · ∇U, ϕ).

Since w ∈

(Ω), from Lemma IX.1.1 and Section III.4.1 we can replace ϕ

with w in the previous identity to obtain

ν| w|

1,2

+ (w · ∇U, w) = −hf, wi− ν(∇U, ∇w) − (U · ∇U, w). (IX.4.57)

We now choose U ≡ V , where V is the extension constructed in Lemma

IX.4. 2. Thus, on the account that Φ ≤ ν/2 and by taking η = ν/4, from

(IX.4.57), Lemma IX.4.2 and Lemma IX.1.1, we ﬁnd

|w|

1,2

≤ C



|f |

−1,2

+ kV k

1,2

+ νkV k

1,2



where C

= C

(n, Ω). Since v

∗

∈ W

1/2,2

(∂Ω), from Lemma IX.4.2, we ﬁnd

that the extension V satisﬁes (IX.4.26). Thus, (IX.4.53) follows from thi s

latter displayed inequality and (IX.4.26).

It remains to show (IX.4. 54). To

this end, we choose U ∈ W

1,2

(Ω) to be the solenoidal extension of v

∗

given in

Exercise III.3.5. From (IX.4.57), Lemma IX.1.1 and the condition (Exercise

III.3.5)

kUk

1,2

≤ c kv

∗

1/2,2(∂Ω)

we then easily obtain

ν|w|

1,2

≤ c

∗

1/2,2(∂Ω)

|w|

1,2



|f |

−1,2

+ νkv

∗

1/2,2(∂Ω)

+kv

∗

1/2,2(∂Ω)



|w|

1,2

and (IX.4 .54) follows from this latter inequality and the assumption on v

∗

Remark IX.4.8 The estimate of generalized solution in terms of the bound-

ary data given in (IX.4.53 ) deserves some comments. We wish to emphasize

that the method we employed, which goes back to J. Leray and E. Hopf (see

the Notes for this Chapter for more details), does not seem to f urnish the es-

timate (IX.4.53) unless we require the boundedness of the set of the boundary

data (that is, v

∗

∈ W

1/2,2

(∂Ω). Moreover, the dependence of the constant c

in (IX.4.53) on the coeﬃcient of kinematic viscosity ν may be very compli-

cated. However, if ν ≥ ν

, for some positive ν

, then c

depends only on ν

These facts seem to have been overlooked by several authors, including myself;

see Finn (19 61a, Theorem 2.3), Ladyzhenskaya (1969, Chapter 5, Section 4),

Galdi (1994b, Theorem VIII.4.1), Finn & Solonnikov (1997, Theorem 3). 

Notice that the constant c

in ( IX.4.53) depends also on ν, because the choice of

η depends on ν, and the constant C in (IX.4.26) depends on η.

IX.4 Existence and Uniqueness with Nonhomogeneous Boundary Data 619

Suﬃcient conditions for the uniqueness of generalized solutions are at once

derived from Theorem IX.2.1 and (IX.4.54), and we ﬁnd the following.

Theorem IX.4.2 The generalized solution v constructed in Theorem IX.4.1

is unique in the class of generalized solutions corresponding to the same f

and v

∗

provided



|f|

−1,2

+ kv

∗

1/2,2(∂Ω)



+ kv

∗

1/2,2(∂Ω)

< C ν,

where C = min{c

/2, 1/

√

k}, while c

, c

and k a re deﬁned in Theorem

IX.4. 1 and in Theorem IX.2.1, respectively.

Remark IX.4.9 If m = 0, that is, if the boundary of Ω is constituted by

only one connected surface (line, for plane ﬂow) Γ , say, condition (IX.4.50) is

automatically satisﬁed, since, by the incompressibility condition,

∗

· n ≡

∂Ω

∗

· n = 0.

Moreover, if m > 1, condition (IX.4.50) furnishes a computable bound on the

ﬂuxes Φ

in terms of ν. It may be of a certain interest to evaluate this bound

when Ω is an annulus. In fact, as we have noticed at the beginning of this

section, such domains cannot admi t, in general, an extension ﬁeld V (α) of v

∗

obeying (IX.4.3) for arbitrary α > 0, and, as a consequence, the Leray-Hopf

construction of steady-state solutions would require identically vanishing Φ

To ﬁx the ideas, take Ω to be the annulus bounded by R and 2R, that is,

Ω =



x ∈ R

: R < |x| < 2R



. (IX.4.58)

Thus, i n the notation of Lemma IX.4.1 and Theorem IX.4.1, we have

d = 2R − R = R, κ

= 1

(x) = −σ

(x) = −∇(log |x|)/2π = −1/(2π|x|),

Ω

1,d



x ∈ R

: R < |x| < 3R/2



Ω

2,d



x ∈ R

: 3R/2 < |x| < 2R



Moreover, we have to give explicit values to the constants κ and c deﬁned

in (IX.4 .31) and (IX.4.3 5), respectively. Concerning κ, from (IX.4.35 ) and

(IX.4.58) we a t once obtain

κ = (3π)

1/4

R/2 ' 1.238

√

However, a sharper estimate can be obtained on κ. Actually, from the La-

dyzhenskaya inequality (II.3.9)

For the value of κ

, see Remark III.6.1.

620 IX Steady Navier–Stokes Flow in Bounded Domains

kuk

≤ 2

−1/4

kuk

1/2

k∇uk

1/2

and from (IX.4.34) we ﬁnd that κ can be estimated by the product of 2

−1/4

times the fourth root of the Poincar´e constant µ, deﬁned a s

µ = µ(Ω) = max

u∈W

1,2

(Ω)

kuk

|u|

1,2

;

cf. (II. 5.3). The value of µ can be calculated from the formula

√

µ = π − 1/(16π) + 163/(3072π

) −93029/(491520π

) + . . . ;

cf. McLachlan (1961,§1.62, eq.(4)). We then recover µ ' 0.10 2 · R

and

κ ' 0.476

√

R. (IX.4.59)

To evaluate the constant c, we observe that, since the products σ

· ∇ψ

i = 1, 2, depend only on r ≡ |x|, the function h in (IX.4.35) depends only

on r. Therefore, a solution b to (IX.4.35) with Ω given in (IX.4.57) can be

chosen of the form

b(x) =

ξh(x)dξ, x ∈ Ω.

Since

∂b

∂x

∂b

∂x

by a direct computation we show that

|b|

1,2

= k∇ · bk

= khk

and we conclude that c = 1 . Collecting all these data and setting Φ ≡ −Φ

, condition (IX.4.5 0) becomes

H|Φ| < ν, (IX.4.60)

where

H ≡

2π



4κ

2π

(A + B) + (C + D)



A =

2π

3R/2

−1

dξ

1/2

, B =

2π

3R/2

−1

dξ

1/2

C =

2π

3R/2

−3

dξ

1/4

, D =

2π

3R/2

−3

dξ

1/4

and κ is gi ven in (IX.4.59). Evaluation of H furnishes

IX.5 Regularity of Generalized Solutions 621

H ' 0.58

and the ﬂux condition (IX.4.60) becomes

|Φ| < 1.72ν.



Remark IX.4.10 The existence result of Theorem IX.4.1 can be extended

to any dim ension n ≥ 4, provided

∗

∈ W

1−1/q,q

(∂Ω), q > n/2,

and

(n)

≡

m+1

i=1



kσ

n/2,Ω

i,d

+ c

kσk

n,Ω

i,d



∗

· n| < ν.

In fact, by Remark IX.4.7 and (IX. 4.56) we obtain

(ν −Φ

(n)

)|u

1,2

≤ (|f|

−1,2

+ ν|V |

1,2

) |u

1,2

+ |(V ·∇V , u

)|.

Moreover, from the H¨older inequality it follows that

|(V ·∇V , u

)| ≤ ckV k

1,2

and since W

1,q

(Ω) ⊂ L

(Ω) for q > n/2 and n ≥ 4 (see Theorem II.3.4), we

then have

|(V · ∇V , u

)| ≤ c

kV k

1,q

1,2

Thus,

(ν − Φ

(n)

)|u

1,2

≤



|f|

−1,2

+ νkV k

1,2

+ c

kV k

1,q



which furnishes the desired uniform bound on |u

1,2

. Concerning the estimate

for the pressure, we refer to Remark IX.3.1. Under the stated assumptions on

the trace norm of v

∗

, estimates similar to (IX.4.53), (IX.4.5 4) continue to

hold for generalized solutions also for n = 4. Therefore, by R emark IX. 2.3,

the uni queness result of Theorem IX.4.2 extends in the same form to n = 4.

For uniqueness i n dimension n ≥ 5, we refer to Remark IX. 5.5. 

IX.5 Regularity of Generalized Sol ut ions

We shall now show certain L

-estimates for weak solutions to problem

(IX.0.1), (IX.0.2). These estimates will imply, in particular, that if Ω and

the data are smooth then the corresponding weak solutio ns are also smooth.

The key tool is a very general result proved in the next Lemma IX.5.1,

regarding a linearized version of problem (IX.0.1 ), (IX.0.2).

622 IX Steady Navier–Stokes Flow in Bounded Domains

Lemma IX.5.1 Let Ω be a bounded domain of R

, n ≥ 2, of class C

u ∈ L

(Ω) wi th ∇ · u = 0 in Ω, in the weak sense, and q ∈ (1, n). Then, fo r

any

F ∈ L

(Ω) , g ∈ W

1,q

(Ω) , w

∗

∈ W

2−1/q,q

(∂Ω) ,

with

Ω

g =

∂Ω

∗

·n ,

the problem

∆w = u · ∇w + ∇π + F

div w = g

)

in Ω

w = w

∗

at ∂Ω

(IX.5.1)

has at least one solution (w, π) ∈ W

2,q

(Ω) × W

1,q

(Ω), that satisﬁes, in addi-

tion, the following estimate

kwk

2,q

+ kπk

1,q

≤ C



kF k

+ kgk

1,q

+ kw

∗

2−1/q,q,∂Ω



, (IX.5.2)

with C = C(n, q, Ω, u). Furthermore, let w ∈ D

1,s

(Ω), for some s ∈ (1, ∞),

satisfy (IX.5.1)

2,3

along with the equati on

(∇w, ∇ϕ) − (u · ∇ϕ, w) = (F , ϕ) , for all ϕ ∈ D(Ω) . (IX.5.3)

Then, if n ≥ 3, necessarily w = w, a .e. in Ω, while, if n = 2, the same

conclusion holds provided s ∈ [2, ∞) and u ∈ L

(Ω), for some q

> 2.

Proof. We begin to show the existence result. In this respect, we cla im that it

is enough to show it with g ≡ 0 and w

∗

≡ 0. Actually, under the assumption

of the lemma, let (w

, π

) ∈ W

2,q

(Ω) ×W

1,q

(Ω) (with π

Ω

= 0 ) be a solution

to the following Stokes problem

∆w

= ∇π

div w

= g

)

in Ω

∂Ω

= w

∗

In view of T heorem IV.6.1, this solution exists (uniquely) and satisﬁes the

estimate

2,q

+ kπ

1,q

≤ C



kgk

1,q

+ kw

∗

2−1/q,q,∂Ω



, (IX.5.4 )

with C = C(n, q, Ω). If we then write the solution to (IX.5.1) in the form

(w = w

+ w

, π = π + π

), we immediately recognize that (w

, π

) solves

(IX.5.1) with g = w

∗

= 0 and with F replaced by F

:= F −u·∇w

. However,

by the H´older and Sobolev inequalities (see Theorem II.3.4), we ﬁnd

ku · ∇w

≤ kuk

k∇w

nq/(n−q)

≤ C kuk

2,q

IX.5 Regularity of Generalized Solutions 623

which ensures F

∈ L

(Ω). Consequently, the above claim follows from this

property and from (IX.5.4). We shall thus prove the lemma when g ≡ 0

and w

∗

≡ 0. By Theorem III.2.1 and Exercise II.2.6, given ε > 0, there are

sequences {v

(j)

} ⊂ C

∞

(Ω), {F

(j)

} ⊂ C

∞

(Ω), and an integer j = j(ε) > 0

such that

ku − u

(j)

+ k∇· u

(j)

+ kF − F

(j)

< ε , for all j ≥ j . (IX.5.5)

Consider the f ollowing sequence of problems

∆w = U

(j)

·∇w + u

(j)

· ∇w + ∇π + F

(j)

∇ · w = 0

)

in Ω

∂Ω

= 0 ,

(IX.5.6)

where U

(j)

:= u

(j)

−u

(j)

. If we formally multiply (IX. 5.6)

by w and integrate

by parts over Ω, we obtai n

|w|

1,2

(∇ · u

(j)

, |w|

) − (F

(j)

, w) . (IX.5.7)

From the embedding T heorem II.3.4 we get

kwk

2n/(n−1)

≤ C

|w|

1,2

where C

= C

(Ω, n), and so, with the help of the H¨older inequality, it f ollows

that

|(∇· u

(j)

, |w|

)| ≤ C

k∇ · u

(j)

|wk

1,2

. (IX.5.8)

Moreover, by the Poincar´e inequality (II.5.1), we ﬁnd

|(F

(j)

, w)| ≤ C

(j)

|w|

1,2

, (IX.5.9)

where C

= C

(Ω). Thus, cho osing ε < 1/C

, from (IX.5.7)–(IX.5.9) it follows

that

|w|

1,2

≤ C

(j)

Thanks to this estimate, we may then use the method employed in the proof

of Theorem IX.3.1 to show, for each ﬁxed j ≥ j, the existence of a generalized

solution (w

(j)

, π

(j)

) ∈ W

1,2

(Ω) × L

(Ω) to (IX.5.6). Let us prove that this

solution is, in fact, more regular and that it satisﬁes, in particular

(j)

, π

(j)

) ∈ W

2,t

(Ω) × W

1,t

(Ω) , for all t ∈ [1, ∞). (IX.5.10)

Let us begin to show that

(j)

, π

(j)

) ∈ W

1,s

(Ω) × L

(Ω), for all s ∈ [1, ∞). (IX.5.11)

Actually, since

(j)

≡ u

(j)

· ∇w

(j)

∈ L

(Ω),

624 IX Steady Navier–Stokes Flow in Bounded Domains

from the results on the Stokes problem established in Theorem IV.6.1 it then

follows that

(j)

, π

(j)

) ∈ W

2,2

(Ω) × W

1,2

(Ω) .

Thus, by the embedding theorem Theorem I I.3.4 we infer that a

(j)

∈ L

(Ω)

with r = 2n/(n−2) if n > 2 and all r > 1 if n = 2. Employing again Theorem

IV.6. 1, we ﬁnd that

(j)

, π

(j)

) ∈ W

2,r

(Ω) × W

1,r

(Ω).

If r ≥ n, that is, n ≤ 4, then (IX.5.1 1) is proved; otherwise,

(j)

∈ L

(Ω), r

= 2n/(n − 4) (> r)

and, again by Theorem IV.6.1 we deduce

(j)

, π

(j)

) ∈ W

2,r

(Ω) × W

1,r

(Ω).

If r

≥ n we arrive at (IX.5.1 1); if not, we iterate the argument as many times

as we please until we derive (IX.5.11). Once (IX.5.11) has been established,

we use the cla ssical results of Theorem IV.6. 1 one more time to prove, in

particular, the validity of (IX.5.10), in any space dimension n ≥ 2. Next, by

means of the H¨older inequality, the embedding Theorem II.3.4, and relations

(IX.5.5) we deduce, for any q ∈ (1, n),

(j)

·∇w

(j)

≤ kU

(j)

k∇w

(j)

nq/(n−q)

+ max

Ω

(j)

|k ∇w

(j)

≤ ε kw

(j)

2,q

+ max

Ω

(j)

|k ∇w

(j)

. (IX.5.12)

In view of (IX.5.10) we can now apply Theorem IV. 6.1 to problem (IX.5.6)

to recover, with q ∈ (1, n),

(j)

2,q

+ kπ

(j)

1,q

≤ C



εkw

(j)

2,q

+ max

Ω

(j)

|k ∇w

(j)

+ kF

(j)



where C

= C

(q, n, Ω). So, taking ε < min{1/C

, 1 /C

}, from the preceding

inequality we derive the following one

(j)

2,q

+kπ

(j)

1,q

≤ C



M(u)kw

(j)

1,q

+kF

(j)



, q ∈ (1, n) , (IX.5.13)

with M(u) ≡ ma x

Ω

(j)

|. Our next task is to prove the existence of a positive

constant C

= C

(q, n, Ω, u), but otherwise indep endent of j, such that

(j)

1,q

≤ C

(j)

. (IX.5.14)

This will be proved by the, by now familiar, contradiction argument. Thus,

assuming the invalidity of (IX.5.14), for any integer m we could ﬁnd {F

(m)

}

IX.5 Regularity of Generalized Solutions 625

such that denoted by {(w

(m)

, π

(m)

)} the corresponding solutions to (IX.5.6)

(with j = m), the inequality

(m)

1,q

> m kF

, for all m ∈ N ,

would hold. By the linearity of the problem, we can take, without loss,

(m)

1,q

= 1, for all m ∈ N, (IX.5.15)

so that the preceding inequality furni shes

(m)

< 1/m , for all m ∈ N . (IX.5. 16)

From the estima te (IX.5.13), we deduce that kw

(m)

2,q

is uniformly bounded

in m, and by Remark II.3.1 and Theorem II.5.2 we can infer the existence of

a ﬁeld W ∈ W

2,q

(Ω) and of a subsequence, ag ain denoted by {w

(m)

}, such

that

(m)

→ W in W

2,q

(Ω)

→ W in W

1,t

(Ω) , for all t ∈ [1, nq/(n −q)).

(IX.5.17)

Furthermore, in view of (IX.5.5 ),

(m)

→ u in L

(Ω). (IX.5.18)

Passing to the limit m → ∞ in (IX.5.6) (with j = m) and using (IX.5.16)–

(IX.5.18), it follows that the limit function W sati sﬁes

(∇W , ∇ϕ) + (u · ∇W , ϕ) = 0, for all ϕ ∈ D(Ω),

∇ · W = 0 , W ∈ W

1,q

(Ω) ∩ W

2,q

(Ω) .

(IX.5.19)

Let us prove that W ≡ 0. If n = 2, since q ∈ (1, 2) by the embedding Theorem

II.3.4 we infer W ∈ H

(Ω), with r

= 2q/(2 − q) > 2, so that, in particular,

W ∈ H

(Ω). Let {ϕ

} ⊂ D(Ω) with ϕ

→ W in H

(Ω), and thus, in

particular, in H

(Ω). This ensures that

lim

k→∞

(∇W , ∇ϕ

) = |W |

1,2

Moreover, for any s ∈ (r

, 2), we have u ∈ L

(Ω) with ∇·u = 0, and so, using

the properties of W and ϕ

along with the embedding Theorem II.3.4, with

the help of Exercise IX.2.1 we ﬁnd

lim

k→∞

(u · ∇W , ϕ

) = (u · ∇W , W ) = 0 .

We next replace ϕ with ϕ

in (IX.5.19)

, pass to the limit k → ∞ and use

the properties stated in the last two displayed equation to obtain W ≡ 0. On

the other hand, from (IX.5.15) and (IX.5 .17) we also have