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

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

We wish to put the right-hand side o f (XIII.7.39) into a diﬀerent form. To

this end, we recall that v obeys the identity (XIII.5.5). Denoting by ψ

the

“cut-oﬀ” function of Theorem XIII.6 .1, we choose, as test function ψ into

(XIII.5.5) the function Φψ

a. We thus o bta in

Φν(ψ

∇v, ∇a)− Φ(ψ

v · ∇a, v) = −Φ(p, ∇·(ψ

a))

−Φν(∇ v, a ⊗∇ψ

) + Φ(v · a ⊗∇ ψ

, v).

(XIII.7.40)

Using the properties of the function a and recalli ng that |∇ψ

| ≤ MR

−1

with

M independent of R, we readily establish the following relations:

lim

R→∞

(ψ

∇v, ∇a) = (∇v, ∇a)

lim

R→∞

(ψ

v · ∇a, v) = (v · ∇a, v)

lim

R→∞

(∇v, a ⊗ ∇ψ

) = lim

R→∞

(v · a ⊗ ∇ψ

, v) = 0.

(XIII.7.41)

Furthermore, reasoning exactly a s in the proof o f (XIII.6.4), we show

lim

R→∞

(p, ∇ · (ψ

a)) = −(p

−p

−

) (XIII.7.42)

with p

constants deﬁned in Lemma XIII.5.1. Collecting (XIII.7.40)–(XIII.7.42)

we ﬁnd

−Φ(v · ∇a, v) + Φ(v · ∇ a, v) = −p

∗

Φ,

which, in view of (XIII.7.39), proves that v obeys the energy inequali ty. The

theorem is completely proved. ut

Remark XI II.7.2 Existence for arbitrary value of the ﬂux can be established

for a w ide class of domains with outlets Ω

whose cross sections Σ

) become

unbounded for large values of x

. This class certainly includes domains for

which Ω

are b odies of rotation of the type considered in the linear case in

Theorem VI.3.1. The proof of this result, which patterns that just given in

Theorem XIII.7.3 , can b e found in Solonnikov & Pil eckas (1977, Theorem 8);

cf. a lso Ladyzhenskaya & Solonnikov (1980, Theorem 4.1). For existence in

domains of more general types, we refer the reader to the work of Solonnikov

(1983, §3.3). 

Remark XI II.7.3 An existence result of the same type as Theorem XIII.7.3

can be proved also in dimension two; see Galdi, Padula, & Passerini (1995)

and R emark XIII.5.3. 

The remaining part of this section is devoted to analyze the uniqueness of

generalized solutions constructed in Theorem XIII.7.3. To reach this goal, we

need an intermediate result that ensures the existence of generalized solutions

to (XII I.5.2) enjoying the additional property of being members of L

(Ω).

This result will be obtained under the assumption that the m a gnitude of the

ﬂux Φ is suitably restricted.

XIII.7 Aperture Domain: Existence and Uniqueness 947

Lemma XIII.7.2 Let c be the constant entering the estimate given in The-

orem VI.5.1. Then, if

|Φ| <

3ν

16c

problem (XIII.5.2) has at least one generalized solution v that satisﬁes

v ∈

1,3/2

(Ω) ∩ L

(Ω).

Moreover, the following estimate holds:

kvk

≤

√

|Φ|. (XIII.7.43)

Finally, denoting by p the pressure ﬁeld corresponding to v according to

Lemma XIII.5 .1, and by p

the constants a ssociated to p by the same lemma,

we have

p − p

∈ L

) ∩ L

3/2

). (XIII.7.4 4)

Proof. Let

X =

1,2

(Ω) ∩

1,3/2

(Ω)

k · k

= | · |

1,2

+ | · |

1,3/2

and set

= {w ∈ X : k · k

≤ δ}.

Clearly, X endowed with the norm k · k

is a Banach space and X

is a closed

subset of X. Consider the mappi ng

L : w ∈ X

→ L(w) = v ∈ X

where v solves

ν(∇v, ∇ϕ) = −(w · ∇w, ϕ), for all ϕ ∈ D(Ω)

= Φ.

(XIII.7.45)

The map L is well deﬁned. Actually, by the Sobolev inequal ity (II.3.7),

|w ·∇w|

−1,3/2

≤ kwk

≤

|w|

1,3/2

. (XI II.7.46)

Moreover, by the interpolation inequality (II.2.7) and again by (II.3.7), for

some θ ∈ (0, 1),

|w·∇w|

−1,2

≤ kwk

2θ

kwk

2(1−θ)

≤

2(1 −θ)

|w|

1,3/2

|w|

1,2

. (XIII.7.47)

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

Thus,

w · ∇w ∈

−1,2

(Ω) ∩

−1,3/2

(Ω),

and from Theorem VI.5.1 we infer that the solution v to (XI II.7.45) i s in

the space X, which proves that L is well deﬁned. By the estimate given in

Theorem VI.5.1, (XIII.7.46), and (XIII.7.47), we obtain

kvk

≤ c



|Φ| +

3ν

kwk



. (XII I.7.48)

Using this inequality, it is easy to show that, for suitably restricted |Φ| the

map L transforms X

into itself, with δ appropriately chosen. In fact, fo r

w ∈ X

, (XIII.7.48) delivers

kvk

≤



|Φ| +

3ν



and so, choosing

δ = 2c|Φ|, (XIII.7.49)

and recalling the assumption of the lemma, we obtain

kvk

≤ δ





= δ, (XIII.7.50)

thus proving the desired property of L. Furthermore, for all w

, w

∈ X

, in

virtue of Theorem VI.5.1, (XIII.7.46), and (XIII.7.47) it easily fol lows that

kL(w

) − L(w

≤

(kw

−w

+ kw

−w

kwk

)

≤

8cδ

3ν

−w

Since, by assumption,

3ν

δ =

16c

|Φ|

3ν

< 1,

we conclude that L is a contraction operator on X

and, therefore, it admits

a ﬁxed point in X

. The existence of a solution v ∈ X to probl em (XIII.5.2)

is thus established. Moreover, the Sobolev inequality (II.3.7), together with

(XIII.7.49) and (XIII.7.50), implies

kvk

≤

√

|v|

1,3/2

≤

√

|Φ|,

proving (XIII.7.43). Finally, the summability properties of the pressure ﬁeld

are at once established from L emma XIII.5.1 and Theorem VI.5.1. The l emma

is then completely proved. ut

XIII.8 Aperture Domain: Summability Properti es of Generalized Solutions 949

Combining Lemma XIII.7.1 with Theorem XIII.6.2 we obtain the following

uniqueness result.

Theorem XIII.7.4 Assume that Φ veriﬁes the condition

|Φ| < mν,

with m = min{3/4c, 3 /16c

} and where c i s the constant introduced in Lemma

XIII.7.1. Then the corresponding generalized solution v constructed in The-

orem XIII.7.3 is unique in the class of g eneralized solutions w satisfying the

energy i nequality (XIII.6.6).

Other sim ple but interesting consequences of Lemma XIII.7.1 and Theorem

XIII.6.2, in light of Theorem XIII.7.4, are given in the following corollary.

Corollary XIII.7.1 Let the a ssumption of Theorem XIII.7.4 be satisﬁed.

Then every generalized solution v corresponding to Φ and verifying the in-

equality (XIII.6.6) satisﬁes the following summability condi tions

v ∈ L

(Ω) ∩

1,3/2

(Ω).

In particular, v satisﬁes the energy equation.

XIII.8 Global Summability of Generalized Solutions for

Flow in an Aperture Domain

The remaining part of this chapter is devoted to the investigation of the

asymptotic structure of generalized soluti ons. As in the case of ﬂows in exte-

rior regions, this study will be performed in two diﬀerent steps. In the ﬁrst,

we determine general summability properties of weak solutions of the type

constructed in Theorem XIII.7.3. Successively, using these condi tions, we wi ll

furnish a complete representation of the solution at large spatial distances.

However, as we proved in Theorem XIII.7.4, in order for a weak solution to

verify the conditions stated in Theorem XIII.7.3, a small value of the ﬂux Φ is

needed. As a consequence, the asymptoti c structure of generalized solutions

corresponding to arbitrary values of Φ remains open.

In this section we shall determine the summability properties o f weak

solutions corresponding to ﬂuxes of suitably restricted size. Such a result will

be achieved as a corollary to a more general one, which we are going to derive.

Notation. Unless the contrary is explicitly stated, in the sequel we shall

denote by Ω the half -space R

Let ϕ ∈ C

(R) be a noni ncreasing nonnegati ve function wi th ϕ(t) = 1

when t ≤ 1 and ϕ(t) = 0 when t ≥ 2. For a > 0 set

(x) = ϕ



|x|



. (XIII.8.1)

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

Clearly, the support of ∇ϕ

is contained in the domain



x ∈ R

: a < |x| < 2a



and

|∇ϕ

(x)| ≤

, x ∈ R

, (XIII. 8.2)

with c (> 0) independent of a and x. We put

Ω

(a)

= R

∩S

, Ω

= R

∩ B

. (XIII.8.3)

The following result holds.

Lemma XIII.8.1 Let v ∈ L

loc

(Ω) ∩ L

(Ω) with ∇·v = 0.

Then there is a

sequence {u

} of solenoidal

functions in Ω such that

∈ L

(Ω) for a ll s ∈ (3/2, 6], for all k ∈ N

lim

k→∞

− vk

= 0.

(XIII.8.4)

In addition to (XIII.8.4), gi ven ε > 0, the sequence {u

} can be chosen such

that

= u

(1)

+ u

(2)

(1)

< ε

supp (u

(2)

) ⊂ Ω

(2)

q,Ω

≤ kvk

q,Ω

, for all q ∈ (1, 6),

(XIII.8.5)

where the number ρ depends only on ε and v.

Proof. Consider the problem:

∇ · z

= −∇ · (ϕ

v) = −∇ϕ

· v

∈ D

1,r

(Ω)

1,r

≤ ck ∇ϕ

· vk

(XIII.8.6)

From Corollary V.3.1 and the assumption on v we know that there is a sol ution

to problem (XIII.8.6) for any r ∈ (1, 6]. From the Sobolev inequality (II.3.7)

we have (see (XI II.8.3)

)

≤ c|z

1,3/2

≤ ck ∇ϕ

3,Ω

(k)

kvk

3,Ω

(k)

and so, in view of (XIII.8.2),

≤ c

kvk

3,Ω

(k)

(XIII.8.7)

In the generalized sense.

XIII.8 Aperture Domain: Summability Properti es of Generalized Solutions 951

with c

independent of k. Moreover, since ∇ϕ

·v ∈ L

(Ω), 1 < r < 3, for all

k ∈ N, agai n by the Sobolev inequality we deduce that

∈ L

3r/ ( 3−r)

(Ω), 1 < r < 3,

that is,

∈ L

(Ω), for all s ∈ (3/2, ∞). (XIII.8.8)

Given ε > 0, we choose ρ = ρ(ε, v) such that

k(1 − ϕ

)vk

< ε/2 (XIII.8.9)

and set, for all suﬃciently large k,

= ϕ

v + z

(1)

= ϕ

(1 − ϕ

)v + z

(2)

= ϕ

Since for k large enough

= ϕ

it follows that

= u

(1)

+ u

(2)

The ﬁeld u

is solenoidal for all k ∈ N. Furthermore, by (XIII.8.8) and as-

sumption, we deduce (XIII.8.4)

. By (XIII.8.7) and the properties of ϕ

also have

−vk

≤ k(1 −ϕ

)vk

+ kz

→ 0 as k → ∞

and we recover (XIII.8.4)

. Statements (XIII.8.5)

and (XIII.8.5)

are evi dent.

Finally, by (XIII.8.7) and (XIII.8.9) for k large enough, it follows that

(1)

≤ k(1 − ϕ

)vk

+ c

kvk

3,Ω

(k)

≤

= ε.

The lemma is proved. ut

Lemma XIII.8.2 Let v, p be a weak solution to the problem

ν∆v = v · ∇v + ∇p

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω,

where, f or all s ∈ (1, 3/2],

∗

∈ L

(∂Ω)

and

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

hhv

∗

1−1/s,s

, hhv

∗

1/2,2

are ﬁnite.

Then if

v ∈ L

(Ω)

it follows that

v ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω), for all q ∈ (1, 2].

Proof. For simplicity, we shall put ν = 1. Consider the following sequence of

problems

∆w = u

·∇w + ∇π

∇ · w = 0

)

in Ω

w = v

∗

at ∂Ω,

(XIII.8.10)

where {u

} is the sequence of functions associated to v by Lemma XIII.8.1

and corresponding to a certain value of ε, which will be speciﬁed later.

It is

simple to show f or problem (XIII.8.10) the existence of a weak solution w

for each k ∈ N. Actually, we may extend the ﬁeld v

∗

to some U ∈ D

1,2

(Ω);

see Section II.10. U need not be solenoidal, but we may add to it a ﬁeld u

such that

∇ · u = −∇· U

u ∈ D

1,2

(Ω)

|u|

1,2

≤ ck∇ · Uk

The ﬁeld u exi sts by virtue of Corollary V.3.1. Thus,

V = u + U

is a solenoidal extension of v

∗

satisfying, by the trace Theorem II.10.2 and

the property of u, the bound

|V |

1,2

≤ c hhv

∗

1/2,2

. (XIII.8.11 )

We look for a solution to (XIII.8.10) of the form w = z + V ,

where

∆z = u

· ∇z + u

· ∇V − ∆V + ∇π

∇· z = 0

)

in Ω

z = 0 at ∂Ω,

(XIII.8.12)

These are the trace norms at the boundary of functions deﬁned in a half-space;

see Section II.10.

Notice that, since v ∈ D

1,2

(Ω), by the embedding Theorem II.3.4, it follows that

v ∈ L

loc

(Ω) so that the assumptions of Lemma XIII .8.1 are satisﬁed.

For simplicity, we shall drop the subscript k in w

and z

XIII.8 Aperture Domain: Summability Properti es of Generalized Solutions 953

Multiplying (XIII.8.1 2)

by z, integrating by parts over Ω, and using (XIII.8.12)

we formally obtain

− |z|

1,2

= (u

· V , z) + (∇V , ∇z). (XIII.8.13)

Using the H¨older inequality in the terms at the right-hand side of this relati on,

together with the Sobolev inequality (II.3.7), we deduce

|(u

· ∇V , z)| ≤ ku

|V |

1,2

kzk

≤ γku

|V |

1,2

|z|

1,2

|(∇V , ∇z)| ≤ |V |

1,2

|z|

1,2

These inequalities along with (XIII.8.11) and (XIII.8.13), yield

|z|

1,2

≤ c

hhv

∗

1/2,2

and recalling that w = z + V , we also have

|w|

1,2

≤ c

hhv

∗

1/2,2

. (XIII.8.14)

The above bound on z allows us to determine, for all k ∈ N, a generalized

solution to (XIII.8.10) satisfying (XIII .8.14). Since

∈ L

(Ω) for all r ∈ (3/2, 6],

and ∇ · u

= 0, by inequality (II.6.22) we ﬁnd for all q ∈ (6/5, 3/2]

· ∇w|

−1,q

≤ ku

6q/(6−q)

kwk

≤ c

6q/(6−q)

|w|

1,2

< ∞.

Thus, we m ay assert

F ≡ u

· ∇w ∈ D

−1,q

(Ω) for all q ∈ (6/5, 3/2]. (XIII.8.15)

Using (XIII.8.15) and Theorem V.3.3, it follows that

w ∈ D

1,q

(Ω) ∩ L

3q/(3−q)

(Ω) for all q ∈ (6/5, 3/2]. (XIII.8.16)

In view of (XIII.8.16), we may give a diﬀerent estimate for F . Speciﬁcally, by

the H¨older inequality and (XIII.8.5)

2,4

we have for all q ∈ (6/5, 3/2]

|F |

−1,q

≡ |u

· ∇w|

−1,q

≤ ku

(1)

kwk

3q/(3−q)

+ kvk

4,Ω

kwk

4q/(4−q),Ω

≤ εkwk

3q/(3−q)

+ kvk

4,Ω

kwk

4q/(4−q),Ω

(XIII.8.17)

From (XIII.8.17 ) and estima te (V.3.28) we deduce

(1 − εc)kwk

3q/(3−q)

+ |w|

1,q

+ kπk

≤ c



hhv

∗

1−1/q,q

+kvk

4,Ω

kwk

4q/(4−q),Ω



(XIII.8.18)

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

where c denotes the constant entering (V.3.28). Thus, choosing ε < 1/c, from

(XIII.8.14) and (XIII.8.18) we deduce, for all q ∈ (6/5, 3/2], that

|w|

1,2

+ kwk

3q/(3−q)

+ |w|

1,q

+ kπk

≤ c



hhv

∗

1/2,2

+ hhv

∗

1−1/q,q

+kwk

4q/(4−q),Ω



(XIII.8.19)

Our next objective is to show the existence of a constant c

= c

(ρ, q, v)

independent of k such that

kwk

4q/(4−q),Ω

≤ c



hhv

∗

1/2,2

+ hhv

∗

1−1/q,q



. (XI II.8.20)

Contradicting (XIII.8.20) means that there are sequences

}, {v

∗m

}

such that if {w

, π

} are weak solutions to the problem

∆w

= u

· ∇w

+ ∇π

∇ · w

= 0

)

in Ω

= v

∗

at ∂Ω,

(XIII.8.21)

we have that

hhv

∗m

1/2,2

+ hhv

∗m

1−1/q,q

→ 0 as m → ∞, (XIII.8.22)

while

4q/(4−q),Ω

= 1. (XIII. 8.23)

In view of (XIII.8.19 ) it follows, i n particular, that

1,2

+ kw

3q/(3−q)

+ |w

1,q

+ kπ

≤ M

with M

independent of m. Therefore, by Theorem I I.1.3, Exercise II.6.2, and

Theorem II.5.2 there is w, π such that (at least along a subsequence)

→ w in

1,q

(Ω) ∩ L

3q/(3−q)

(Ω) ∩

1,2

(Ω)

→ w in L

(Ω

) for all r ∈ (1, 3q/(3 − q)), R > 0

→ π in L

(Ω).

(XIII.8.24)

Because of (XIII.8.22) and the trace Theorem I I.10.1, it follows that w has

zero trace at ∂Ω; also, ∇ · w = 0 and so

w ∈ D

1,q

(Ω) ∩ D

1,2

(Ω). (XIII.8.25)

Moreover, owing to (XIII.8.5)

2,4

, we have

XIII.8 Aperture Domain: Summability Properti es of Generalized Solutions 955

kuk

≤ M

, (XIII.8.26)

with M

independent of m, and we may select a subsequence (denoted again

by {u

}) and ﬁnd a ﬁeld U ∈ L

(Ω) such that

→ U weakly in L

(Ω). (XIII.8 .27)

By (XIII.8.2 4), for all ψ ∈ C

∞

(Ω), we have

(∇w

, ∇ψ) → (∇w, ∇ψ)

(π

, ∇ · ψ) → (π, ∇ · ψ).

(XIII.8.28)

Also, by (XIII.8.27), it follows that

· ∇ψ, w

) − (U · ∇ψ, w) = ((u

− U ) · ∇ψ, w)

−(u

· ∇ψ, (w − w

))

≡ I

(m) + I

(m).

(XIII.8.29)

By the H¨older inequality,

k∇ψ · wk

3/2

≤ k∇ψk

q/( q−1)

kwk

3q/(3−q)

and (XIII.8.24) and (XIII.8.27) imply

lim

m→∞

(m) = 0. (XI II.8.30)

In addition,

|(u

·∇ψ, (w −w

))| ≤ ku

kψk

kw − w

s,%

where % = supp (ψ) and s < 3q/(3 − q). Thus, by (XIII.8.24), we ﬁnd

lim

m→∞

(m) = 0. (XI II.8.31)

Collecting (XIII.8.28)–(XIII.8.31 ), taking into account that w

, π

is a gen-

eralized solution to (XIII.8.21), and usi ng (XIII.8.22), it follows that

0 = lim

m→∞

{(∇w

, ∇ψ) − (u

·∇ψ, w

) − (π

, ∇· ψ)}

= (∇w, ∇ψ) − (U · ∇ψ, w) −(π, ∇ · ψ).

Therefore, by (XIII.8.25), w solves the homogeneous problem

(∇w, ∇ψ) = (U · ∇ψ, w) + (π, ∇ · ψ) for all ψ ∈ C

∞

(Ω)

w ∈ D

1,q

(Ω) ∩ D

1,2

(Ω).

(XIII.8.32)

We may take, in particular, ψ ∈ D(Ω) to obtain