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

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546 VIII Steady Generalized Oseen Flow in Exterior Domains

with c

= c

(ρ, B), which, in turn, by virtue of (VIII.4.58), produces

ess sup

t≥0

[]∇w(t)[]

3/2,R,B

2ρ

≤ c

khk

∞,R

∞

, (VIII.4.70)

with c

= c

(ρ, B). If x ∈ B

2ρ

, we have x −y| ≤ 3ρ, and so, choosing this time

β = 3ρR in (VII I .3.12)

, we deduce

w(x, t)| ≤ c

khk

∞,R

∞

|x −y|

≤ c

, |x| ≤ 2ρ .

Combining this latter with (VIII.4.70) we obtain the desired result for ∇w.

In order to estimate D

w, we start with the representation (VIII.4.67) and

|x| ≤ 2ρ. Choosing β = 3ρR in (VIII.3.12)

, we obtain, as before,

w(x, t)| ≤ c

k∇hk

∞,R

∞

|x −y|

≤ c

, |x| ≤ 2ρ . (VIII.4.71)

If |x| ≥ 2ρ, we can diﬀerentiate the fundam ental tensor solution twice with

respect to x a nd use (VIII.3.12)

with β = ρR. We thus obtain

w(x, t)| ≤ c

khk

∞,R

∞

|x − y|

(1 + 2Rs(x − y))

, |x| ≥ 2 ρ ,

with c

= c

(ρ, B), which, in turn, by (VIII.4.58), implies

ess sup

t≥0

[]D

w(t)[]

2,R,B

2ρ

≤ c

khk

∞,R

∞

The statement about D

w then follows from this inequality and (VIII.4.71).

The theorem is completely proved. ut

VIII.5 Existence, Uniqueness, and Pointwise Estimates

of Solutions in the Whole Space

The objective of this section is to prove existence, uniqueness, and corre-

sponding estimates of solutions v, p to the inhomogeneous generalized Oseen

problem

∆v + R

∂v

∂x

+ T (e

× x · ∇v − e

×v) = ∇p + f

∇ · v = 0











in R

lim

|x|→∞

v(x) = 0 ,

(VIII.5.1)

under suitable assumptions on the datum f .

We will keep the notation used in the previous section given, in particular,

in (VIII.4.46).

Our main result is the following.

VIII.5 Existence, Uniqueness, and Pointwise Estimates in the Whole Space 547

Theorem VIII.5.1 Let F be a second-order tensor ﬁeld such that ∇ · F ∈

), with []F[]

2,R

< ∞, and l et g ∈ L

), q ∈ (3, ∞), with suppo rt

contained in Ω

, for some ρ > 0. Then, the problem (VIII.5.1) with f ≡

∇· F + g has at least one solution such that

v ∈ W

2,2

loc

) ∩ D

2,2

) ∩ D

1,2

), |]v|]

1,R

< ∞,

p ∈ W

1,2

loc

) ∩D

1,2

) ∩ L

), r > 3/2.

(VIII.5.2)

This solution satisﬁes the estimates

|v|

2,2

+ |v|

1,2

+ |p|

1,2

≤ C

(k∇ · Fk

+ []F[]

2,R

+ kgk

)

[|v|]

1,R

+ kpk

≤ C

([]F[]

2,R

+ kgk

)

(VIII.5.3)

where C

= C

(ρ, q, B), C

= C

(ρ, r, q, B) whenever R, T ∈ [0, B], for som e

B > 0.

Moreover, assume, in particular, that F = 0 and g satisﬁes the further

assumption g ∈ W

1,∞

(Ω

). Then, v satisﬁes also

[]∇v[]

+ []D

v[]

< ∞ if R = 0 , (VIII.5.4)

[]∇v[]

3/2,R

+ []D

v[]

2,R

< ∞ if R 6= 0 , (VIII.5.5)

and there are constants C

= C

(ρ), C

(ρ, B) if R ∈ (0, B] such that

[]∇v[]

+ []D

v[]

≤ C

kgk

1,∞

, if R = 0 , (VIII.5.6)

[]∇v[]

3/2,R

+ []D

v[]

2,R

≤ C

kgk

1,∞

, if R 6= 0 . (VIII.5.7)

Finally, if (v

, p

) is a generalized solution to (VIII.5.1) corresponding to the

same f, then v = v

and p = p

+ const.

Proof. The uniqueness part is an immediate consequence of Theorem VIII.2.1.

Moreover, from Theorem VIII.1.1, Theorem VIII. 1.2, and Lemma VIII.3.2 it

follows that the only property for v that remains to be proved is the asymp-

totic properties of v, al ong with the corresponding estimates.

In order to

prove these la tter, we consider the following unsteady Cauchy problem asso-

ciated with (VIII.5 .1):

∂u

∂t

= ∆u + R

∂u

∂x

−T (e

× u − e

× x · ∇u)

−∇q − ∇ ·F − g

∇ · u = 0











in R

× (0, ∞)

u(x, 0) = 0, x ∈ R

(VIII.5.8)

Recall the notation given in (VIII.4.46).

Notice that ∇ · F ∈ D

−1,2

) and that |∇ · F |

−1,2

≤ c []F[]

2,R

. Likewise, since

|(g, ϕ)| ≤ kgk

6/5

kϕk

≤ c kgk

|ϕ|

1,2

for all ϕ ∈ D

1,2

(Ω), we deduce |g|

−1,2

≤ c kgk

548 VIII Steady Generalized Oseen Flow in Exterior Domains

We b egin by proving the properties stated in (VIII.5.2 ) and (VIII.5.3). To this

end we shall prove that (i) []u(t)[]

1,R

is uniformly bounded in time, and that (ii)

u(·, t) converges as t → ∞ to the solution v of (VIII.5.1) in appropriate norms.

As a consequence, the asymptotic (spatial) behavior o f v will be shown to be

the same as that o f u. To reach this goal, we make a change of variables that

brings (VIII.5.8) into an appropriate (unsteady) Oseen problem. Speciﬁcally,

we deﬁne

χ = Q(t) · x,

w(χ, t) = Q(t) · u(Q

(t) · χ, t), π(χ, t) = q(Q

(t) · χ, t) ,

G(χ, t) = −Q(t) · F (Q

(t) · χ) · Q

(t) , h(χ, t) = −Q(t) · g(Q

(t) · χ) ,

(VIII.5.9)

where

denotes transpo se, and Q = Q(t), t ≥ 0, is a one-parameter family

of second-order tensors satisfying the following initial-value problem:







= T Q · W (e

) ,

Q(0) = I ,

(VIII.5.10)

with

W (e

) =





0 0 0

0 0 −1

0 1 0





. (VIII.5.11)

Since the matrix W is skew-symmetric, it follows that for each t ≥ 0, Q(t)

deﬁnes a proper orthogonal transformation, i.e.,

Q(t) · Q(t)

= Q

(t) · Q(t) = I (I = identity tensor) .

More precisely, by integrating (VII I .5.10))–(VII I .5.11), we obtain

Q(t) =





1 0 0

0 cos(T t) −sin(T t)

0 sin(T t) cos(T t)





. (VIII.5.12)

We also notice that

W (e

) · a = e

× a , for all a ∈ R

, (VIII.5.13)

and moreover, that

Q(t) · e

= e

, for all t ≥ 0. (VIII.5.14)

Now, using (VIII.5.9), (VIII.5.10), and (VIII.5.13) together with the identity

· Q(t) = −Q

(t) ·

we obtain



•

≡



VIII.5 Existence, Uniqueness, and Pointwise Estimates in the Whole Space 549

∂w

∂t

= Q(t) ·



∂u

∂t



•

(t) · Q(t) · x



·∇u + Q

(t)·

•

Q (t) · u



= Q(t) ·



∂u

∂t

−T (e

× x · ∇u − e

× u)



(VIII.5.15)

and

∆

w = Q(t) · ∆u. (VIII.5.16)

Moreover, from (VIII.5.9) and (VIII.5.14) we also obtain

∂w

∂χ

(χ, t) = Q(t) ·



(t) · e

) ·∇u(x, t)



= Q(t) ·

∂u

∂x

(x, t) . (VIII.5.17)

Therefore, coll ecting (VIII.5.15)–(VIII.5.17), we deduce that the Cauchy prob-

lem (VIII.5.8) can be equivalently rewritten as fol lows:

∂w

∂t

= ∆w + R

∂w

∂χ

− ∇π + ∇ · G + h

∇ · w = 0











in R

× (0, ∞) ,

w(χ, 0) = 0 , χ ∈ R

(VIII.5.18)

At this point we observe that in view of the assumptions on F and the or-

thogonality properties of the family Q(t), t ≥ 0,

ess sup

t≥0

[]G(t)[]

2,R

= []F[]

2,R

ess sup

t≥0

k∇· G(t)k

= k∇· Fk

ess sup

t≥0

kh(t)k

= kgk

As a consequence, from Theorem VIII.4.4 it f ollows tha t problem (VI II.5.18)

has one and only one solution (w, φ) such that

(w, ∇φ) ∈ L

) , φ ∈ L

2,6

) , (VIII.5.19)

which, in addition, satisﬁes the following estimate, for all r ∈ (3/2, ∞):

ess sup

t≥0

[]w(t)[]

1,R

+ ess sup

t≥0

kφ(t)k

≤ C ([]F[]

2,R

+ kgk

) , (VIII.5.20)

with C depending on (r, q and) an upper bound for R and T . We now go back

to the original ﬁelds u, q. Since Q = Q(t) is o rthog onal, we have the identities

|x| = |χ|, |u(x, t)| = |w(χ, t)|, |q(x, t)| = |φ(χ, t)|, (VIII.5.21)

and so from (VIII.5.20), we obtain

550 VIII Steady Generalized Oseen Flow in Exterior Domains

ess sup

t≥0

[]u(t)[]

1,R

+ ess sup

t≥0

kq(t)k

≤ C ([]F[]

2,R

+ kgk

) . (VIII.5.22 )

It remains to prove the convergence of u to v when t → ∞. Setting

V (χ, t) = Q(t) · v(Q

(t) · χ), P (χ, t) = p(Q

(t) ·χ) ,

U(χ, t) = w(χ, t) − V (χ, t) , π(χ, t) = φ(χ, t) − P (χ, t) ,

and observing that (see (VIII.5.15))

∂V

∂t

= −T Q(t) · (e

×x · ∇

v − e

×v) , (VIII.5.23)

we ﬁnd that the pair (U , π) satisﬁes

∂U

∂t

= ∆U + R

∂U

∂χ

−∇π

∇· U = 0











in R

× (0, ∞) ,

U(χ, 0) = v(χ), χ ∈ R

(VIII.5.24)

Multiplying both sides of the ﬁrst equation by ∇ψ, with ψ ∈ C

∞

), we

easily show, f or a.a. t ∈ [0, T], that

(∇π(t), ∇ψ)

= 0 , for all ψ ∈ C

∞

) ,

which implies that π(t) is harmonic in the whole o f R

, for a.a. t ∈ [0, T ].

Furthermore, the result of Lemma VIII.2.2 and (VIII.5.20) implies π(t) ∈

), for a.a. t ∈ [0, T ]. Therefore, by Exercise II.11. 11, we obtain π(x, t) =

0 f or all x ∈ R

and a.a. t ∈ [0, T], and (VIII.5.24) becomes

∂U

∂t

= ∆U + R

∂U

∂χ

∇ · U = 0











in R

× (0, ∞) ,

U(χ, 0) = v(χ), χ ∈ R

(VIII.5.25)

We now observe that the initi al datum v is in L

) and that moreover, by

the properties of w and v (see (VIII.5.20), (VIII.5.19), and (VIII.5.23)), we

have

V ∈ L

) ,

∂V

∂t

∈ L

loc

((0, T] ×R

) , D

V ∈ L

) ,

w ∈ L

) ,

∂w

∂t

, D

w ∈ L

) .

From these conditions we infer, in particular,

U ∈ L

) ,

∂U

∂t

, D

U ∈ L

loc

((0, T ] × R

) . (VIII.5.26)

VIII.5 Existence, Uniqueness, and Pointwise Estimates in the Whole Space 551

Thus, from Theorem VIII.4.3 we deduce

kU k

≤ Ct

−1/4+3/(2σ)

kvk

, σ > 6 ,

k∇Uk

≤ Ct

−1/2

kvk

for all t > 0. From these two latter relations with σ = 12, and with the help

of inequality (II.9.7), we obtain

|U(x, t)| ≤ Ct

−1/8

kvk

, for all t ≥ 1 ,

namely

|u(x, t) − v(x, t)| ≤ Ct

−1/8

kvk

, for all t ≥ 1 , (VIII.5.27)

since |U| = |u −v|. We are now in a position to obtain the desired asymptotic

behavior a nd the corresponding estimate f or v. In fact, by (VIII.5.22) and

(VIII.5.27), we obtain, for all t ≥ 1,

|v(x)|(1 + |x|)(1 + 2Rs(x)) ≤ |u(x, t) − v(x)| (1 + |x|)(1 + 2Rs(x))

+|u(x, t)|(1 + |x|)(1 + 2Rs(x))

≤ c

(1 + |x|)(1 + 2 Rs(x))t

−1/8

kvk

+ c

[]F[]

2,R

Passing to the limit t → ∞, we ﬁnall y deduce

|v(x)|(1 + |x|)(1 + 2Rs(x)) ≤ c

[]F[]

2,R

which furnishes the desired result for v. In order to complete the proof of

the ﬁrst part of the theorem, it remains to prove the stated properties for

the pressure along with the estimate (VIII.5.3 ) . From Theorem VIII.1.1 and

the assumptions on F we recover p ∈ W

1,2

loc

). This property together with

(VIII.2.5) implies, possibly by modifying p by the addition of a constant,

p ∈ D

1,2

) ∩ L

). Then, from (VIII.5.1), (VIII.5.2)

, we deduce that

×x ·∇v − e

×v) ∈ L

) ,

which, coupled with (VIII.2.26), furnishes

× x · ∇v − e

× v) ∈ H(R

) .

We next operate with the Helmholtz–Weyl projection operator P (see Remark

III.1.1) on both sides of (VIII.5.1)

. Observing that

P ∆v = ∆v , P



∂v

∂x



∂v

∂x

P (e

× x · ∇v − e

× v) = e

× x · ∇v − e

× v ,

it follows that

552 VIII Steady Generalized Oseen Flow in Exterior Domains

T (e

×x ·∇v − e

×v) = −R

∂v

∂x

−∆v + P ( ∇ · F + g) .

Therefore, recalling that v sati sﬁes (VIII.5.3), we obtain

T ke

× x · ∇v − e

× vk

≤ c ([]F[]

2,R

+ k∇ · F k

+ kgk

) , (VIII.5.28)

where c depends on B. Employing (VIII.5.28), (VIII.5.1 )

, and the estimate

(VIII.5.3) for v, we thus conclude that

|p|

1,2

≤ c

([]F[]

2,R

+ k∇· F k

+ kgk

) .

Next, we have to show that p ∈ L

), r ∈ (3/2, ∞), and prove the corre-

sponding estimate given in (VIII.5.3). In fact, these properties can be shown

in an entirely simila r way to that employed at the end of the proof of The-

orem VIII.4.4 to prove analogous properties for the pressure φ, and so the

proof will be only sketched here. We multiply both sides of (VIII.5.1)

by ∇χ,

χ ∈ D

1,2

(Ω), to obtain

(∇p, ∇χ) = −(∇ · F + g, ∇χ) .

Choosing χ as the solution to the Poisson equation ∆χ = ψ, ψ ∈ C

∞

)

and using the facts that p ∈ L

(Ω) and q > 3, one shows that

|(p, ψ)| ≤ c ([]F[]

2,R

+ kgk

) kψk

, for all ψ ∈ C

∞

), and all r

∈ (1, 3) .

From this relation it then follows that p ∈ L

), for all r ∈ (3/2, ∞) and

that kpk

≤ c ([]F[]

2,R

+ kgk

). This concludes the proof of the ﬁrst part of

the theorem. Now assume F = 0 and that g satisﬁes the further assumption

g ∈ W

1,∞

(Ω

). By the orthogonality properties of the family Q(t), t ≥ 0, we

thus have

ess sup

t≥0



kh(t)k

∞,R

∞

+ k∇h(t)k

∞,R

∞



= kgk

1,∞

. (VIII.5.29)

Moreover, from the results of Theorem VIII.4.4 applied to the Cauchy problem

(VIII.5.18) with R = 0 and G = 0, we obtain

ess sup

t≥0



[]∇w(t)[]

+ []D

w(t)[]



≤ c

ess sup

t≥0



kh(t)k

∞,R

∞

+ k∇h(t)k

∞,R

∞



that is, by (VIII.5.29),

ess sup

t≥0



[]∇w(t)[]

+ []D

w(t)[]



≤ c

kgk

1,∞

Bearing in mind (VIII.5 .21), we infer that this inequality continues to hold

with u in place of w. Consequently, reasoning exactly as in the proof of

the ﬁrst part (precisely, the argument following (VIII.5.22)), we establish the

properties (VIII.5.4) and (VIII.5.6) for v. By completely analogous reasoning,

we can prove (VIII.5.5) and (VIII.5.7) as well. The theorem is completely

proved.

VIII.5 Existence, Uniqueness, and Pointwise Estimates in the Whole Space 553

Remark VI II.5.1 In connection with the assumptions of Theorem VIII.5.1,

it may be of some interest to ﬁnd conditions under which a vector ﬁeld f,

deﬁned on a generic exterior do main Ω ⊆ R

, can be written in the form

∇· F , with F satisfying the a ssumptions of Theorem VIII.5.1. For example,

if f is of bounded support in Ω and satisﬁes some further assumptions, then

f can be represented in the desired way; see Exercise II I.3.9 and Exercise

VIII.5.1. When R = 0, a simple suﬃcient condition (but of course, one could

formulate others as well) is that f ∈ L

(Ω), it i s bounded on any bounded

set, and decays to zero as |x| → ∞ like |x|

−3

, as shown in the following.

Lemma VIII.5.1 Let Ω be an exterior domain of R

, and let f ∈ L

(Ω)

with []f[]

< ∞. Then, there exists a second-order tensor ﬁeld F such that

f = ∇ · F (∈ L

(Ω)), and moreover, []F[]

< ∞.

Proof. Extend f to zero outside Ω and set ψ

(x) = (E ∗ f

)(x), i = 1, 2, 3,

where E is the (three-dimensional) Laplace fundamental solution. With the

help of Lemma II.9.1, Lemma II.9.2, and Theorem II.11.2 and employing the

hyp othesis made on f , we can show that ψ

is well deﬁned and that it satisﬁes

∆ψ

= f

, in R

, i = 1, 2, 3 . We next want to estimate the quantity

(x) =

4π

−y

|x −y|

(y) dy , i, k = 1, 2, 3 . (VIII.5.30)

It is easy to see that D

ψ ∈ L

∞

). This is because, by the properties of f,

(x)| ≤ c []f[]

|x−y|<1

|x −y|

|x−y|≥1

|x −y|

(1 + |y|)

≤ c

[]f[]

We now set |x| = R > 0, and write the integral in (VIII.5.30) as the sum of

three integrals, over the domains B

R/2

, B

R/2,2R

, and B

, respectively. We

denote these integrals, in order, by I

(R), I

(R), and I

(R). We have

(R)| ≤

R/2

| ≤ c

kf k

Moreover,

(R)| ≤ c

[]f []

R/2,2R

|x −y|

≤ c

[]f []

dr ≤ c

[]f[]

Finally,

(R)| ≤

|f | ≤ c

kfk

The lemma then fol lows with F

= D

, i, j = 1, 2, 3. ut



554 VIII Steady Generalized Oseen Flow in Exterior Domains

Exercise VIII.5.1 Let Ω ⊂ R

, n ≥ 2 and let f ∈ L

(Ω), q ≥ n, with bounded

support and f

Ω

= 0. Show that there exists a second-order tensor ﬁeld F deﬁned

in Ω, and with bounded support, such that

f = ∇ · F in Ω , kF k

∞

≤ c kf k

Hint: Use Theorem III. 3.1 and Theorem III.3.2.

VIII.6 On the Pointwise Asymptotic Behavior of

Generalized Soluti ons

In this section we will investigate the behavior at large distances from the

boundary of generalized solutions to (VIII.0.2), (VIII.0.7) in a generic (suﬃ-

ciently smooth) exterior domain of R

. Precisely, we shall show that, under

suitable assumptions on the body force, the magnitude of the velocity ﬁeld can

be pointwise bounded by the function w(x) = (1 + |x|)(1 + 2Rs(x)), where,

we recall, s(x) = |x| + x

. Thus, if R > 0, the function w shows the same

asymptotic behavior as the Oseen fundamental solution E (see (VII.3 .23) a nd

Remark VII.3.1), whereas if R = 0 , the behavior is the same as the Stokes

fundamental solution (see (IV.2.6)). We also recall that by virtue of the Mozzi–

Chasles transformation considered in the Introduction, R = 0 if and only if

the translational velocity v

and the ang ular velocity ω are orthogonal, or,

in particular, v

= 0.

Therefore, w(x) does not present a wave-like behavior

unless v

has a nonzero comp onent in the direction of the angular velocity.

Also in this section, we will keep the notation introduced in (VIII.4.46) .

In order to prove the above-mentioned properties, we need, as usual, some

preparatory results.

Lemma VIII.6.1 Let Ω be an exterior domain of class C

, and let R, T ∈

[0, B], for some B > 0. Assume that f ∈ L

(Ω

), R > δ(Ω

), and v

∗

∈

2−1/q,q

(∂Ω), q ∈ (1, ∞). Then, the g eneralized solution (v, p) to (VIII.0.2),

(VIII.0.7) satisﬁes

v ∈ W

2,q

(Ω

), p ∈ W

1,q

(Ω

) , (VIII.6.1)

for a ll r ∈ (δ(Ω

), R). Moreover,

kvk

2,q,Ω

+ kpk

1,q,Ω

≤ C( kfk

q,Ω

+ kv

∗

2−1/q,q,∂Ω

+ kvk

q,Ω

+ kpk

q,Ω

(VIII.6.2)

where C = C(Ω, r, R, B).

Proof. The proof of (VIII.6.1) i s a consequence of Theorem VIII.1.1 and the

assumptions made on the data F and v

∗

. The proof of (VIII.6.2) goes as

follows. We write (VIII.0.7) as a Stokes problem

We recall t hat we are assuming ω 6= 0; otherwise, the analysis coincides with that

performed in Chapter VII.

VIII.6 On the Pointwise Asymptotic Behavior of Generalized Solutions 555

∆v = ∇p + H

∇ · v = 0

)

in Ω ,

v = v

∗

, at ∂Ω ,

with

H = −R

∂v

∂x

−T (e

×x ·∇v − e

×v) + f.

Exploiting Theorem IV.5.1 and Theorem IV.5.3, and taking into account this

expression for the function H, we readily get

kvk

2,q,Ω

+ kpk

1,q,Ω

≤ C (kfk

q,Ω

+ kv

∗

2−1/q,q,∂Ω

+kHk

−1,q,Ω

+ kvk

q,Ω

+ kpk

q,Ω

)

with C depending only on Ω, r, and R. Since

kHk

−1,q,Ω

≤ C(1 + R + T )kvk

q,Ω

+ kfk

q,Ω

we obtain the desired estimate. ut

We are now in a position to establish the main result of this section.

Theorem VIII.6.1 Let Ω be an exterior domain of class C

. Assume that F

is a second-order tensor ﬁeld on Ω such that ∇·F ∈ L

(Ω) with []F[]

2,R

< ∞,

and that v

∗

∈ W

3/2,2

(∂Ω). Let v be the corresponding generalized soluti on

to (VIII.0 .2), (VIII.0.7) with f ≡ ∇ · F. Then, denoting by p the pressure

ﬁeld associated to v by Lemma VIII.1.1, we have

v ∈ W

2,2

loc

(Ω) ∩ D

1,2

(Ω) ∩ D

2,2

(Ω) , []v[]

1,R

< ∞,

p ∈ D

1,2

(Ω) ∩ L

(Ω

) for all q

∈ (3/2, 6], and all q

∈ (6, ∞) ,

(VIII.6.3)

where ρ is an arbitrary number greater than δ(Ω

). Moreover, v, p satisfy the

following estimate

|v|

2,2

+ |v|

1,2

+ []v[]

1,R

+ |p|

1,2

+ kpk

,Ω

≤ C(k∇·F k

+ []F[]

2,R

+ kv

∗

3/2,2,∂Ω

) ,

(VIII.6.4)

where C depends only on Ω, B, q

, and ρ, whenever R, T ∈ [0, B].

Proof. In view of the stated assumptions, Lemma VIII.6.1, Lemma VIII.2.1,

and Lemma VIII.2.2, we have only to show that []v[]

1,R

< ∞, p ∈ L

(Ω) ∩

(Ω

), along with the validi ty of the corresponding estimates. Notice tha t

by virtue of Lemma VIII.6.1, Lemma VIII.2 .2, and the assumptions on F, we

can suppose p ∈ L

(Ω).

To reach the above goal, for a ﬁxed R > δ(Ω

), we denote by ϕ a smooth

“cut-oﬀ” function that is 0 for |x| ≤ R and is 1 for |x| ≥ 2R, and set ϑ = 1−ϕ.

We then introduce the ﬁelds