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

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496 VIII Steady Generalized Oseen Flow in Exterior Domains

lim

|x|→∞

v(x) = 0 . (VIII.0.2)

We recall that in (VIII.0.1), v

and ω are given constant vectors representing

the translational and angula r velocity, respectively, of the rigid motion of B.

We shall assume that Ω is an exterior domain of R

, and refer the reader to

the Notes for this chapter for some remarks and relevant bibliography related

to the two-dimensional case.

In order to describe problems and results, it is convenient to rewrite

(VIII.0.1) and (VIII.0.2) in a suitable dimensionless form . To this end, we

assume, without loss, that ω is directed along the posi tive x

-axis, that is,

ω = ω e

, while v

= v

e, v

≥ 0.

Moreover, we scale the length with

d = δ(Ω

), and the veloci ty with v

, if v

6= 0, and with ω d otherwise. There-

fore, intro ducing the dimensionless numbers

(Reynolds number) T =

ωd

(Taylor number) , (VIII.0.3)

the system (VIII.0.1) assumes the following form

∆v + R

e · ∇v + T (e

× x · ∇v − e

× v) = ∇p + f

∇ · v = 0

)

in Ω ,

v = v

∗

at ∂Ω,

(VIII.0.4)

where now v, v

∗

, p, and f are nondimensional quantities.

If Ω ≡ R

the

above choice of d is no longer possible, but we can still give a meaning to

(VIII.0.4), which is what we shall do hereinafter.

At thi s point we observe that, in general, ω and v

, that is, e

and e, have

diﬀerent directions. However, by shifting the coordinate system by a constant

quantity, we can always reduce the original equations to new ones where e =

. This change of coordinates, known as Mozzi–Chasles transformation (see

Mozzi 1763, Chasles 1830), reads as follows:

∗

= x − λ e

×e , λ :=

≡

ω d

. (VIII.0.5)

Thus, deﬁning

Ω

∗



∗

∈ R

: x

∗

= x − λe

×e , for some x ∈ Ω



∗

) = v(x

∗

+ λe

×e) , p

∗

) = p(x

∗

+ λe

× e) ,

∗

) = f (x

∗

+ λe

×e) ,

R = R

e · e

(VIII.0.6)

Of course, we suppose ω 6= 0; otherwise, the analysis reduces to that already

performed in Chapter V, when v

= 0, and in Chapter VII, when v

6= 0.

Notice that, in contrast to the O seen problem studied in the previous chapter,

here t he body force has been scaled in such a way that dimensionless body force

is −f instead of −R

f . This is because we may allow R

= 0, while keeping a

nonzero force term.

VIII Introduction 497

the system (VIII.0.4) becomes (with stars omitted)

∆v + R

∂v

∂x

+ T (e

× x · ∇v − e

× v) = ∇p + f

∇ · v = 0











in Ω

v = v

∗

at ∂Ω,

(VIII.0.7)

Throughout this chapter, we shall therefore focus on the resolution of

problem (VIII.0 .7), (VI II.0.2).

We would like to po int out the following interesting feature of (VIII.0 .7).

In view of what we have found in the previous chapter, we may guess that

the wake-like behavior of the velocity ﬁeld v at large distances is produced

by the term R

∂v

∂x

. Now, this term is zero whenever the “eﬀective” Reynolds

number R vanishes. In vi ew of the Mozzi-Chasles transformation, this happens

not only when (as intuitively expected) v

= 0, but, more generally, when

·ω = 0, namely, when the translational velocity of the body is perpendicular

to its angular velocity. In fact, as we shall prove later on, the formation of a

“wake” is possible, i n a suitable sense, if and only if v

· ω 6= 0; see Section

VIII.6. For a physical interpretation of this circumstance, we refer to the

Introduction to Chapter XI.

The study of the mathematical properties of the solutions to (VIII.0.7),

(VIII.0.2) is, i n principle, much more challenging than the analogous study

performed for the Oseen problem (VII.0.1), (VII.0.2), the main reason being

the presence, in (VIII.0.7)

, of the term e

× x · ∇v, whose coeﬃcient be-

comes unbounded as |x| → ∞. One i mportant consequence of this fact is that

(VIII.0.7) can by no means be viewed as a perturbation of (VII.0.1), even for

“small” T (“small” angular velocities, that is).

Despite this diﬃculty, one is able to prove, with relative ease, the exi stence

of at least one generalized solution to (VIII.0.7), (VIII.0.2) and to show that

such a solution is smooth, provided the data are equally smooth; see Section

VIII.1. As in the case of the Oseen approximation, the generalized solution

is constructed via the Galerkin method, with the help of a suitabl e basis and

of an appropriate a priori estimate for the Dirichlet norm of v; see Theorem

VIII.1.2. This estimate can be established thanks to the crucial fact that (as

the reader will immediately verify)

Ω

× x · ∇ϕ · ϕ − e

×ϕ · ϕ) = 0 , for all ϕ ∈ D(Ω) .

With a view to appl ications to the full nonlinear problem, as in the case of

the Oseen approximation, also in the case at hand the next important question

to i nvestigate is the behavior at la rge distances of generalized solutions. This

problem already arises, naturally, in the study of the uniqueness of generalized

solutions, which is here established by a method diﬀerent from those used in

Theorem V.2.1 a nd Theorem VII.1.2 for the Stokes and Oseen approximation,

498 VIII Steady Generalized Oseen Flow in Exterior Domains

respectively. In fact, this method relies chieﬂy on a sharp result concerning

the asymptotic behavior of the pressure ﬁeld associated with a generalized

solution corresponding to a body force that is square-summable in Ω

, for

suﬃciently large R; see Section VIII .2.

Unlike the analogous problems for the Stokes and Oseen approximations

analyzed in Section V.3 and Section VII.6, the study of the asymptotic be-

havior of generalized solutions to (VIII.0.7), (VII I.0.2) does not appear to be

feasible by means of the classical method based on volume potential represen-

tations along with asymptotic estimates of the fundamental solution. Actually,

based on heuristic considerations, we are expecting that the velocity ﬁeld de-

cays li ke |x|

−1

, uniformly fo r large |x|. However, as shown by Farwig, Hishida

& M¨uller (2004, Propositi on 2.1), see also Hishida (2006, Proposition 4.1),

the fundamental tensor solution, G = G(x, y), associated with the equatio ns

(VIII.0.1)

(with Ω ≡ R

) does not satisfy a uniform estimate of the type

|G(x, y)| ≤ C|x − y|

−1

, for all x, y ∈ R

with C independent of x, y, at least in the case R = 0, which is, in fact, the

most complicated.

Therefore, we will argue in a diﬀerent way.

The approach we shall follow to study the asymptotic behavior of gener-

alized solutions and to show the corresponding estimates is treated in Section

VIII.3 through Section VIII.6. It was ﬁrst introduced by Galdi (200 3) and

then further generalized a nd improved by G aldi & Silvestre (2007a, 2007b).

It develops according to the following steps. In the ﬁrst step, by means of a

“cut-oﬀ” technique that we have already used in previous chapters, we reduce

the original problem to an analogous one in the whole space. At this stage, the

above-mentioned uniqueness property plays a fundamental role, because we

can then identify the generalized solution in the whole space problem with the

original one, for suﬃciently large |x|, x ∈ Ω. In the second step, we consider

the time-dependent version of (VIII.0.7) in the whole-space (Cauchy problem)

corresponding to the sam e “body force” and to zero initial data; see (VIII.5.8).

We thus show that at each time t ≥ 0, the velocity ﬁeld, u = u(x, t), of the

corresponding solution decays at least like |x|

−1

for large |x|, and that it is

uniformly bounded in time by a function that exhibits the same spatial decay

properties as the Stokes or Oseen fundamental tensor, depending on whether

R is zero or not zero, respectively. Therefore, in this sense, u shows a “wake-

like” feature whenever R 6= 0. Successively, thanks to these pointwise, uniform

(in time) estimates, we may thus pass to the limit t → ∞ and show, on the one

hand, that u(x, t) converges, uniforml y pointwise, to the generali zed solutio n

v = v(x) of the steady-state problem, and that on the other hand, v has the

same asymptotic properties and that i t obeys the same estimates as u does.

In the last two sections of the chapter, we will investigate the summability

properties of generalized solutio ns in homogeneous Sob olev spaces, together

with corresponding estimates, when f belongs to the Lebesgue space L

, for

suitable values of q, and v

∗

is i n the appropriate trace space. As we shall see,

VIII.1 Generalized Solutions. Regularity and Existence 499

these results are, formally, completely anal ogous to those shown in Theorem

VII.7.1 for the Oseen problem with m = 0 and n = 3.

Unless otherw ise stated, throughout this chapter we s hall always assume

T > 0,

whereas we take R ≥ 0.

VIII.1 Generalized Solutions. Regularity and Exi stence

We begin with a weak formulation of the generalized Oseen problem through

a, by now, familiar procedure. Thus, multi plying (VIII.0.7)

by ϕ ∈ D(Ω)

and integrating by parts, we obtain

(∇v, ∇ϕ) −R



∂v

∂x

, ϕ



−T (e

×x ·∇v −e

×v, ϕ) = −[f, ϕ]. (VIII.1 .1)

Deﬁnition VIII.1.1. A vector ﬁeld v : Ω → R

is called a q-weak (or q-

generalized) solution to (VII I.0.2), (VIII. 0.7) if for some q ∈ (1, ∞),

(i) v ∈ D

1,q

(Ω);

(ii) v is (weakly) divergence-free in Ω;

(iii) v assumes the value v

∗

at ∂Ω (in the trace sense) o r, if the veloci ty at

the boundary is zero, v ∈ D

1,q

(Ω);

(iv) lim

|x|→∞

|v(x)| = 0;

(v) v satisﬁes (VIII.1.1) for all ϕ ∈ D( Ω).

As usual, if q = 2, v will be called simply a w eak (or generalized) solution to

(VIII.0.2), (VIII.0 .7).

Remark VI II.1.1 If v is a q-weak solution then by Lemma II.6.1, v ∈

1,q

loc

(Ω), and if Ω is locally Lipschitz, then v ∈ W

1,q

loc

(Ω). Concerning (iii),

see Remark V.1.1. Mo reover, if q ∈ (1, 3), then, by (iv) and Theorem II.6.1(i),

it follows that v ∈ L

3q/(3−q)

(Ω), and that

kvk

3q/(3−q)

≤ c |v|

1,q

where c = c(q, Ω). Final ly, in regards to (iv), we notice that, if f is in L

(Ω

for a suﬃciently large ρ then it can be shown that every generalized solution

tends to zero as |x| → ∞ , uniformly pointwise; see Exercise VI II.2.1. 

If the function f has a suﬃcient degree of regularity, to each q-weak so-

lution we can associate a corresponding pressure ﬁeld in a way completely

analogous to that used in Lemma IV.1.1. Speciﬁcally, we have the fol lowing

result, whose proof we leave to the reader as an exercise.

See footnote 1.

500 VIII Steady Generalized Oseen Flow in Exterior Domains

Lemma VIII.1.1 Let Ω be an exterior domain in R

, n ≥ 2. Suppose f ∈

−1,q

(Ω

), 1 < q < ∞, for a ny bounded subdomain Ω

, with Ω

⊂ Ω. Then

to every q-weak solution v we can associate a pressure ﬁeld p ∈ L

loc

(Ω) such

that

(∇v, ∇ψ) − R



∂v

∂x

, ψ



−T (e

× x · ∇v − e

× v, ψ) = (p, ∇ · ψ) − [f, ψ]

(VIII.1.2)

for all ψ ∈ C

∞

(Ω). Furthermore, if Ω is locally Lipschitz and f ∈

−1,q

(Ω

), R > δ(Ω

), then p ∈ L

(Ω

Remark VI II.1.2 The result stated in Lemma VIII.1.1 for Ω locally Lips-

chitz is weaker than its counterpart for the Sto kes problem, proved in Lemma

V.1. 1, for reasons that are analogous to that explained in Remark VII.1.2 for

the Oseen problem. Basically, this is due to the f act that the functional

(∇v, ∇ψ) − R



∂v

∂x

, ψ



−T (e

×x ·∇v − e

×v, ψ) + [f, ψ]

is not continuous in ψ ∈ D

1,q

(Ω) if we merely require v ∈ D

1,q

(Ω). In other

words, under this assumption alone on v we cannot guarantee the existence

of a constant c = c(v, R, T ) such that





∂v

∂x

, ψ



+ T (e

×x ·∇v − e

×v, ψ)



≤ c |ψ|

1,q

, for all ψ ∈ C

∞

(Ω) ,

and so, we cannot apply Corollary III.5.1, but only i ts weaker version given

in Corollary III.5.2. 

The following result furnishes regularity of q-weak solutions.

Theorem VIII.1.1 Let f ∈ W

m,q

loc

(Ω), m ≥ 0, 1 < q < ∞, and let

v ∈ W

1,q

loc

(Ω), p ∈ L

loc

(Ω),

with v weakly divergence-free, satisfy (VIII.1.2) for all ψ ∈ C

∞

(Ω). Then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω).

In particular, if f ∈ C

∞

(Ω), then v, p ∈ C

∞

(Ω). Furthermore, if Ω is o f class

m+2

and

Actually, these assumptions are satisﬁed by any q-weak solution. In fact, they are

implied by the following one:

v ∈ L

loc

(Ω) , ∇v ∈ L

loc

(Ω), with v satisfying (VI II.1.1) for all ϕ ∈ D(Ω).

For under this hypothesis, by Lemma II.6.1, v ∈ W

1,q

loc

(Ω) and then, by Lemma

VIII.1.1, we deduce p ∈ L

loc

(Ω); see also Remark VIII.1.2.

VIII.1 Generalized Solutions. Regularity and Existence 501

f ∈ W

m,q

loc

(Ω), v

∗

∈ W

m+2−1/q,q

(∂Ω),

then

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω) ,

provided v ∈ W

1,q

loc

(Ω).

In particular, if Ω is of class C

∞

, and f ∈ C

∞

(Ω),

∗

∈ C

∞

(∂Ω), then v, p ∈ C

∞

(Ω

), for all bounded Ω

⊂ Ω.

Proof. The proof follows at once from Theorem IV.4.1 and Theorem IV.5.1 if

one bears in m ind that (VIII.1.2) can be viewed as a weak form of the Stokes

equation with f replaced by f − R

∂v

∂x

−T (e

×x ·∇v − e

×v). ut

We shall next establi sh the existence of generali zed solutions. Precisely, we

have the fol lowing.

Theorem VIII.1.2 Let Ω be a three-dimensional exterior, locally Lipschitzian

domain. Given

f ∈ D

−1,2

(Ω), v

∗

∈ W

1/2,2

(∂Ω) ,

there exists at least one generali zed solution to (VIII.0.2), (VIII.0.7 ). This

solution satisﬁes the estimates

kvk

2,Ω

+ |v|

1,2

≤ c



|f |

−1,2

+ (1 + R + T )kv

∗

1/2,2(∂Ω)



|v(x)| = o (1/

|x|) as |x| → ∞,

kpk

2,Ω

/ R

≤ c

{|f|

−1,2

+ (1 + R + T )|v|

1,2

(VIII.1.3)

for all R > δ(Ω

). In (VIII.1.3) p is the pressure ﬁeld associated to v by

Lemma VIII.1.1, whil e c

= c

(R, Ω) (c

→ ∞ as R → ∞).

Proof. The method of proof is very close to that of Theorem VII.2.1. We look

for a solution of the f orm

v = w + V

+ σ, (VIII.1.4)

where V

and σ are solenoidal ﬁelds introduced in the proof of Theorem

VII.2.1, which we will recall here for the reader’s sake. Speciﬁcally, V

∈

1,2

(Ω) is the solenoidal extension of v

∗

− σ|

∂Ω

of bounded support in Ω

constructed in the proof o f Theorem V.1.1, while with the o rigin of co ordinates

◦

Ω

σ =

4π

∇



|x|



Φ =

∂Ω

∗

·n .

(VIII.1.5)

Notice that any q-generalized solution enjoys this requirement, as a consequence

of the stated assumptions on Ω and Remark VIII.1.2.

See ( IV.6.1) for the deﬁnition of the norm involving p.

502 VIII Steady Generalized Oseen Flow in Exterior Domains

Thus,

1,2

≤ c kv

∗

1/2,2(∂Ω)

σ = O(1/|x|

2+|α|

) , |α| = 0, 1, as |x| → ∞.

(VIII.1.6)

As a consequence, w is required to be a member of D

1,2

(Ω) and to satisfy,

for a ll ϕ ∈ D(Ω), the equation

(∇w, ∇ϕ) − R



∂w

∂x

, ϕ



− T (e

×x · ∇w − e

× w, ϕ)

= −[f , ϕ] − (∇V

, ∇ϕ) + R



∂V

∂x

, ϕ



+ T (e

×x · ∇V

−e

×V

, ϕ),

(VIII.1.7)

where we set

≡ V

+ σ.

However, we have the identity

× x · ∇σ − e

× σ = 0 . (VIII.1.8)

In fact, denoting by ε

ijk

the alternating symbol

and by L the left-hand side

of (VIII. 1.8) times 4π/Φ, we obtain (with r ≡ |x| and i = 1, 2, 3)

= ε

k1`

−

) + ε

i1m

= 3 ε

213

+ 3 ε

312

= 0 .

Using (VIII.1.8) in (VIII.1.7), we deduce

(∇w, ∇ϕ) − R



∂w

∂x

, ϕ



− T (e

×x · ∇w − e

×w, ϕ)

= −[f , ϕ] − (∇V

, ∇ϕ) + R



∂V

∂x

, ϕ



+ T (e

× x · ∇V

− e

×V

, ϕ) .

(VIII.1.9)

It is clear that, provided we show the existence of a function w ∈ D

1,2

(Ω)

satisfying (VIII.1.9) for all ϕ ∈ D(Ω), the ﬁeld (VIII.1.4) sati sﬁes all require-

ments of a generalized sol ution to (VIII.0.2), (VIII.0.7) a s given in Deﬁni-

tion VIII.1.1. In fact, (VIII.1.6)

, and the properties of V

and w, imply

v ∈ D

1,2

(Ω). In addition, v is solenoida l and assumes the value v

∗

at the

boundary. Finally, in vi ew of (VIII.1.6)

, we obtain

|v(x)| ≤

|w(x)|+ O(1/|x|

) , (VIII.1.10)

We recall that the alternating symbol is deﬁned as fol lows: ε

ijk

= 1 if (i, j, k)

is an even permutation of (1,2,3); ε

ijk

= −1 if (i, j, k) is an odd permutation of

(1,2,3), and ε

ijk

= 0 if any two of i, j, k are equal.

VIII.1 Generalized Solutions. Regularity and Existence 503

which, by Lemma II.6.2, delivers (VIII.1.3)

. Thus, to establish the theorem

it remains to prove the existence of the ﬁeld w and the validity of estimates

(VIII.1.3)

1,3

. To this end, let {ϕ

} be the base of D

1,2

(Ω) given in Lemma

VII.2.1. We shal l construct an “approximate solution” w

to (VII I.1.9) in

the following way:

`=1

(∇w

, ∇ϕ

) −R



∂w

∂x

, ϕ



− T (e

×x · ∇w

−e

× w

, ϕ

)

= −[f , ϕ

]−(∇V

, ∇ϕ

)+R



∂V

∂x

, ϕ



+T (e

× x · ∇V

−e

×V

, ϕ

)

≡ F

, k = 1, 2, . . ., m.

(VIII.1.11)

Using (ii) of Lemma VII.2. 1 we obtain

`=1

(ξ

− ξ

) = F

, k = 1, 2, . . ., m (VIII.1.12)

where

≡ R



∂ϕ

∂x

, ϕ



+ T (e

×x · ∇ϕ

−e

×ϕ

, ϕ

System (VIII.1 .12) is linear in the unknowns ξ

, ` = 1, . . . , m, and since

= −A

, we deduce that the determinant of the coeﬃcients is nonzero.

As a consequence, for each m ∈ N, system (VIII.1.12) admits a uniq uely

determined solution. Let us multiply (VIII.1.12) by ξ

and sum over k from

1 to m. We obtain

1,2

= −[f , w

]

−(∇V

, ∇w

)+R



∂V

∂x

, w



+T (e

× x · ∇V

− e

×V

, w

(VIII.1.13)

By (VIII.1.6 )

, and the fact that f ∈ D

−1,2

(Ω), we readily show that

−[f, w

] ≤ |f|

−1,2

1,2

−(∇V

, ∇w

) ≤ c

∗

1/2,2(∂Ω)

1,2

−



+ σ,

∂w

∂x



≤ c

∗

1/2,2(∂Ω)

1,2

× x · ∇V

− e

×V

, w

) ≤ c

∗

1/2,2(∂Ω)

1,2

(VIII.1.14)

Recalling (VIII.1.14) and using (VI II.1.13) we obtain

1,2

≤ c



|f |

−1,2

+ (1 + R + T )kv

∗

1/2,2(∂Ω)



. (VIII.1.1 5)

504 VIII Steady Generalized Oseen Flow in Exterior Domains

Therefore, the sequence {w

} remains uniformly bounded in D

1,2

(Ω), and

by Exercise II.6.2, there exist a subsequence, denoted again by {w

}, and a

function w ∈ D

1,2

(Ω) such that in the limit m → ∞,

(∇w

, ∇ϕ) → (∇w, ∇ϕ), for all ϕ ∈ D

1,2

(Ω). (VI II.1.16)

Also, by (VIII.1.15) and Theorem II.2.4 we infer

|w|

1,2

≤ c



|f|

−1,2

+ (1 + R + T )kv

∗

1/2,2(∂Ω)



(VIII.1.17)

with c = c(Ω). Furthermore, from Exercise II.5.8 and by a simple diagona liza-

tion procedure, we can select another subsequence, again denoted by {w

}

such that

→ w in L

(Ω

), for all R > δ(Ω

). (VIII.1.18)

For ﬁxed k, we then pass to the limit m → ∞ into (VIII.1.11)

to deduce

with the help of (VIII.1.16), (VIII.1.18 ) that v satisﬁes (VIII.1.1) for all ϕ

Since, by Lemma VII.2.1, every ϕ ∈ D(Ω) can be approximated by a linear

combination of ϕ

in the W

1,2

-norm and in the norm k ρ · k

, ρ = (1 +

|x|), we easily establish the validity of (VIII.1.1) for all ϕ ∈ D(Ω). Let us

next prove estimates (VIII.1.3)

1,3

. From (VIII.1 .3)

, Theorem II.6.1 , and the

H¨older inequality, we deduce

kvk

2,Ω

≤ |Ω

1/3

kvk

6,Ω

≤ c |Ω

1/3

|v|

1,2

, (VIII.1.19)

where c = c(Ω), and so, since

|v|

1,2

≤ |w|

1,2

+ |V

1,2

+ |σ|

1,2

inequality (VIII.1.3)

follows from (VIII.1.17), (VIII.1.19), and (VIII.1 .6). Let

us ﬁnally prove (VIII.1.3)

. For ﬁxed R > δ(Ω

), we add to the pressure p

(deﬁned through Lemma VIII.1.1 ) the constant

C(R) = −

|Ω

Ω

p ,

so that

Ω

(p + C) = 0.

Successively, we take ψ into (VII.1 .2) as a solution to the problem

∇ · ψ = p + C in Ω

ψ ∈ W

1,2

(Ω

) ,

kψk

1,2

≤ c

kp + Ck

2,Ω

for som e c

= c

(Ω

). This problem is resolvable by vi rtue of Theorem II.4.1

and so from (VIII.1.2) and the Schwarz inequality we have

VIII.2 Generalized Solutions. Uniqueness 505

kp + Ck

2,Ω

≤ c

(|v|

1,2

+ (R+ T )kvk

2,Ω

+ |f|

−1,2

) (VII I.1.20)

which, by (VIII.1.3)

, in turn implies (VIII.1.3)

. The theorem is thus com-

pletely proved. ut

VIII.2 Generalized Solutions. Unique ness

The objective of this section is to prove that the generalized solution deter-

mined in Theorem VIII.1.2 is unique in the class of all generalized solutions

corresponding to the same data. In view of the linearity of the problem, this

amounts to showing that the only generalized solution to (VIII.0.2), (VIII.0.7)

corresponding to v

∗

≡ f ≡ 0 is identically zero. As in the case of the Oseen

problem, in the case at hand, a fortiori, we are not allowed to replace v f or ϕ

in (VIII.1.1) if v merely belong s to D

1,2

(Ω); see Remark VIII.1.2. Moreover,

even if we could, it would not be obvious to conclude that the contribution

from the second and third term on the left-hand side of (VIII.1.1) with ϕ ≡ v

is zero. We will, therefore, argue diﬀerently. To this end, we need to prove a

number of preparatory results.

Lemma VIII.2.1 Let v be a generalized soluti on to (VIII.0.2), (VIII.0.7)

corresponding to f ∈ L

(Ω

), for some ρ > δ(Ω

). Then v ∈ D

2,2

(Ω

), for all

r > ρ. Moreover, there is c = c(r, B) (with c(r) → ∞ as r → ρ

) such that

for a ll R, T ∈ [0, B],

2,Ω

≤ c (kfk

2,Ω

+ |v|

1,2

) . (VIII.2.1)

Finally, if Ω = R

and f ∈ L

), then v ∈ D

2,2

) and the following

estimate holds:

2,R

≤ c



kf k

2,R

+ |v|

1,2



, (VIII.2.2)

with c

= c

(B).

Proof. In view of the assumption on f and Theorem VIII.1.1, we k now that

v ∈ W

2,2

loc

(Ω

), p ∈ W

1,2

loc

(Ω

) and that they satisfy the following equations:

∆v + R

∂v

∂x

+ T (e

× x · ∇v − e

× v) = ∇p + f

∇ · v = 0











a.e. in Ω

(VIII.2.3)

Let R > r > ρ, and denote by ψ

= ψ

(|x|) a smooth, non-negative “cut-

oﬀ” function that is 0 for |x| ≤ ρ and |x| ≥ 2R, while it is 1 for |x| ∈

[r, R]. Moreover, we may take |D

|, |D

(

√

)| ≤ M R

−|α|

, all |α| ≥ 0,

with M independent of R. We next dot-multiply both sides of (VIII.2.3)