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

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766 XI Three-Dimensional Flow in Exterior Domains. Rotational Case

we m ay choose η such that η +

|Φ|

4πr

< 1/R, and from (XI.3.11)–(XI.3.13) and

Lemma IX.3.2 we may conclude that the algebraic system (XI.3.10) has a t

least one solution for every m ∈ N. Moreover, from (XI.3.10)–(XI.3.13) it also

follows that



1 − R



η +

|Φ|

4πr



1,2

≤ C [|V + v

∞

1,2

(1 + RkV + v

∞

)

+R|V + v

∞

1,6/5

+ T kV

1,6/5

+ |f|

−1,2

] ,

(XI.3.15)

which shows that under the conditi on (XI.3.14), the sequence {u

} is (uni-

formly) bounded in D

1,2

(Ω). Thus, we can select a subsequence, denoted again

by {u

}, such that as m → ∞,

→ u in D

1,2

(Ω). (XI.3.16)

This property along with Theorem II.1.3(i) and (XI.3.15) implies



1 − R



η +

|Φ|

4πr



|u|

1,2

≤ C [|V + v

∞

1,2

(1 + RkV + v

∞

)

+R|V + v

∞

1,6/5

+ T kV

1,6/5

+ |f|

−1,2

] ,

(XI.3.17)

where C = C(Ω). Furthermore, since D

1,2

(Ω) ,→ W

1,2

(Ω

) for all R >

δ(Ω

) (see Lemma II.6.1 ), from Exercise I I.5.8 and the Cantor diagonalization

argument we secure the existence of another subsequence, still called {u

such that as m → ∞,

→ u in L

(Ω

), q ∈ [1, 6) , (XI.3.18)

for any R > δ(Ω

). At this point, we may employ a rguments analogous to

those used in the proof of Theorem X.4.1 (in pa rti cular, the argument follow-

ing (X.4.29)) to show that the ﬁeld u satisﬁes (XI.3. 10) with u

≡ u, for all

k ∈ N. The ﬁnal step is to replace, in this resulting equation, ϕ

with an arbi-

trary ϕ ∈ D(Ω). Thi s goal is again achieved by the same procedure employed

in the proof of Theorem X.4.1, thanks to the properties of the linear hull of

the basis {ϕ

}; see Lemma VII.2.1 . The only term that deserves a littl e more

attention is the “rotational term” (e

×x·∇u, ϕ

)−(e

×u, ϕ

). However, the

replacement of ϕ

with ϕ ∈ D(Ω) in this latter is carried out exactly as de-

scribed in the proof of Theorem VIII .1.2, after equation (VIII.1.19). We may

then safely conclude that u satisﬁes (XI.3.9), for all ϕ ∈ D(Ω), that is, the

ﬁeld v := u + V is a generali zed solution to (XI.0. 10), (XI.0.11). Moreover,

we prove (see (X.4.34)) that v + v

∞

satisﬁes (XI.3.4). Finally, choosing in

(XI.1.4) the f unction ψ introduced in (X.4.38) and proceeding as in (X.4.39),

we easily establish the validity of (XI.3.4). We shall next establish the validity

of the energy inequality (XI.3.5). We begin by o bserving that from (XI.3.10)

and (XI.3 .11) it follows that

XI.3 Existence of Generalized Solutions 767

1,2

= −(∇(V + v

∞

), ∇ u

) − R[(u

· ∇(V + v

∞

), u

)

+((V + v

∞

) · ∇(V + v

∞

), u

) − (e

· ∇(V + v

∞

), u

)]

−T (e

× V

− e

×x · ∇V

, u

) − [f , u

] .

(XI.3.19)

We would like to let m → ∞ in this relation (along a subsequence if necessary).

Obviously, in vi ew of (XI.3.16) and (XI.3.1 8), and the assumption on f , and

recalling that V

is of bounded support, we obta in

lim

m→∞

{T (e

×V

− e

× x · ∇V

, u

) − [f, u

]}

= T (e

×V

−e

× x · ∇V

, u) − [f, u]

lim

m→∞

(∇(V + v

∞

), ∇ u

) = (∇(V + v

∞

), ∇ u) .

(XI.3.20)

Furthermore, by exactly the same procedure used in the proof of Theorem

X.4. 1 to prove the vali dity of the generalized energy inequality in the irrota-

tional case (see the argument fol lowing (X.4.40)), we obtain

lim

m→∞

{(u

· ∇(V + v

∞

), u

) + ((V + v

∞

) · ∇(V + v

∞

), u

)}

= (u · ∇(V + v

∞

), u) + ((V + v

∞

) · ∇(V + v

∞

), u)

lim

m→∞

· ∇(V + v

∞

), u

) = (e

· ∇(V + v

∞

), u) .

(XI.3.21)

Thus, passing to the limit m → ∞ in (XI.3.19), from (XI.3.2 0)–(XI.3.21) and

Theorem II.1.3(i), we conclude that

|u|

1,2

+(∇(V + v

∞

), ∇ u) + R[(u · ∇(V + v

∞

), u)

+((V + v

∞

) · ∇(V + v

∞

), u) − (e

· ∇(V + v

∞

), u)]

+T (e

× V

− e

×x · ∇V

, u) + [f, u] ≤ 0 .

(XI.3.22)

We next observe that since u = v −V ,

|u|

1,2

+ (∇(V + v

∞

), ∇ u) = |v + v

∞

1,2

− (∇(V + v

∞

), ∇ (v + v

∞

))

= 2



kD(v)k

− (D(v), D(V ))



(XI.3.23)

where in the last step, we have used the identity g iven in footnote 4 of Section

X.2. Moreover,

(u·∇(V +v

∞

), u)+((V +v

∞

)·∇(V +v

∞

), u) = ((v+ v

∞

)·∇(V +v

∞

), v−V ) .

(XI.3.24)

Thus, collecting (XI.3.22)–(XI.3 .24), we deduce that

2kD(v)k

−2(D(v), D(V )) + R[((v + v

∞

) · ∇(V + v

∞

), v −V )

−(e

· ∇(V + v

∞

), v − V )] + T (e

× V

− e

×x · ∇V

, v − V )

+[f, v − V ] ≤ 0 .

(XI.3.25)

768 XI Three-Dimensional Flow in Exterior Domains. Rotational Case

We now notice that in view of the assumptions made on f and v

∗

, by Theorem

XI.1. 2 we have, on the o ne hand, that (w := v + v

∞

, p) ∈ W

2,2

(Ω

) ×

1,2

(Ω

), for all ρ > δ(Ω

), and, on the other hand, that (w, ep), with ep given

in (XI.1.6), satisﬁes (XI.2.4), that is,

∇ · T (w, ep) + R(v

∞

− w) · ∇w − T e

×w = f

∇ · w = 0

)

a.e. in Ω

∂Ω

= v

∗

+ v

∞

(XI.3.26)

Let ψ

= ψ

(|x|) be a smooth, nonincreasing “cut-oﬀ” function that is 0 for

|x| ≤ R and is 1 for |x| ≥ 2R, R > δ(Ω

), with |∇ψ

(x)| ≤ M/R, where M

is a constant independent of R and x. If we then dot-multiply both sides of

(XI.3.26)

by ψ

(V + v

∞

) and integrate by parts over Ω, we obtain

∂Ω

n · T (w, ep) · (v

∗

+ v

∞

) − 2(D(v), D(ψ

(V + v

∞

))) − (ep ∇ψ

, V + v

∞

)

−R



(ψ

w · ∇w, V + v

∞



∂ψ

∂x

w, V + v

∞



+(ψ

∞

· ∇(V + v

∞

), w)



+T (ψ

×(V + v

∞

), w) + R

∂Ω

∗

+ v

∞

)

∞

· n = [f , V + v

∞

] ,

(XI.3.27)

where we used e

×x ·∇ψ

(x) = 0 for all x ∈ Ω. In (XI.3.27) we now employ

the form (XI.3.1) of the extension V along with its summability properties

and the identity (XI.3.8) for σ, and recall that, by Theorem XI.1.1 and the

assumption on f, ep ∈ L

(Ω) + L

(Ω). Thus, letting R → ∞, it is not hard to

show that (XI.3.27) leads to

−

∂Ω

n · T (w, ep) ·(v

∗

+ v

∞

) + 2(D(v), D(V ))

+ R[(w · ∇w, V + v

∞

) + (e

· ∇(V + v

∞

), w)]

− T (e

×V

−e

× x · ∇V

, w) −R

∂Ω

∗

+ v

∞

)

∞

· n

+ [f, V + v

∞

] = 0 .

(XI.3.28)

Summing side by si de (XI.3.25) and (XI.3.28), and recalling that w = v +v

∞

we obtain

2kD(v)k

−

∂Ω

n · T (w, ep) · (v

∗

+ v

∞

) − R

∂Ω

∗

+ v

∞

)

∞

· n

+[f, v + v

∞

] + R[(w · ∇(V + v

∞

), w) + (w ·∇w, V + v

∞

)

+(e

· ∇(V + v

∞

), V + v

∞

)]

−T (e

× V

− e

×x · ∇V

, V + v

∞

) ≤ 0 .

(XI.3.29)

XI.3 Existence of Generalized Solutions 769

By an easily justiﬁed integration by parts that uses the summability properties

of w and V + v

∞

, we obtain

(w ·∇(V + v

∞

), w) + (w · ∇w, V + v

∞

) =

∂Ω

∗

+ v

∞

)

∗

+ v

∞

) ·n .

(XI.3.30)

Likewise,

·∇(V + v

∞

), V + v

∞

) =

∂Ω

∗

+ v

∞

)

·n . (XI.3.31)

Finally, recalling that V + v

∞

= V

+ Φσ (see (XI.3.1)), we obtain

−(e

× V

−e

×x · ∇V

, V + v

∞

)

= (e

×·∇V

, V

) + Φ(e

× x · ∇σ − e

× σ, V

)

+Φ

∂Ω

· σ)e

×x · n

= Φ

∂Ω

· σ)e

×x ·n +

∂Ω

× x · n .

(XI.3.32)

We next observe that for any ﬁxed r > δ(Ω

), we have

∂Ω

|σ|

×x ·n =

Ω

∇ · (| σ|

× x) = 2

Ω

× x · ∇σ · σ = 0 ,

because e

×x ·∇σ(x) ·σ(x) = 0, for all x ∈ Ω. Therefore, (XI.3.32) delivers

−(e

×V

−e

×x ·∇V

, V +v

∞

) =

∂Ω

∗

∞

)

×x ·n . (XI.3.33)

Placing (XI.3.30), (XI.3.31), and (XI.3.3 3) into (XI.3.2 9), we then conclude

that

2kD(v)k

−

∂Ω



n·T ( w, ep)·(v

∗

∞

)−

∗

∞

)

∗

·n



+[f , v+v

∞

] ≤ 0 ,

namely, the validity of the energy inequality. It remains to show the estimate of

the solution in terms of the data. Thus, assuming |Φ| ≤ 2πr

/R, and choosing

(for example) η = 1/(4R), (XI.3.17) furnishes

|u|

1,2

≤ 4[C |V + v

∞

1,2

(1 + RkV + v

∞

)

+R|V + v

∞

1,6/5

+ T kV

1,6/5

+ |f |

−1,2

] .

(XI.3.34)

If we now use in (XI.3.34) the inequalities (X.4.6) (with v

∞

≡ v

∞

), and recall

that u = (v +v

∞

)−(V + v

∞

), we obtain (XI.3.6), which com pletes the proof

of the theorem. ut

Remark XI.3.1 For future reference, we observe that in the energy inequal-

ity (XI.3.5), if, in addi tion, f ∈ L

6/5

(Ω),we can then replace [f , v + v

∞

] with

(f, v + v

∞

). 

770 XI Three-Dimensional Flow in Exterior Domains. Rotational Case

XI.4 Global Summability of General ized Solutions when

· ω 6= 0

As in the irrotational case, the fundamental step in deriving the asymptotic

structure of generalized solutions is to prove sui table summability properties

at large distances for the velocity ﬁeld v and the pressure p, much more

detailed than those available at the outset, namely, (v + v

∞

) ∈ D

1,2

(Ω) ∩

(Ω), with p satisfying the conditions given in Theorem XI.1.2.

The proof of the desired properties will be a chieved by following the same

arguments used in the irrotational case, with only the foresight to replace

Theorem VII.7.1 with Theorem VIII.8.1. As a matter of fact, once this basic

replacement is made, the proof is essentially the same.

Speciﬁcally, we have the following result

Theorem XI.4.1 Let Ω be a C

-smooth exterior three-dimensional domain,

and assume, for some q ∈ (1, 2), that

f ∈ L

(Ω) ∩ L

3/2

(Ω) , v

∗

∈ W

2−1/q,q

(∂Ω) ∩ W

4/3,3/2

(∂Ω) . (XI.4.1)

Then, every generalized solution v to the Navier–Stokes problem (XI.0.10)–

(XI.0.11) corresponding to f , v

∗

and to v

·ω 6= 0, and the associated pressure

ﬁeld

p, satisfy (v + v

∞

, p) ∈ X

(Ω), with X

(Ω) deﬁned in (X.6.5).

Proof. Since their actual value is completely irrelevant, throughout the proof

we set, for simplicity, R = T = 1. Moreover, we set u = v + v

∞

. Since

u ∈ D

1,2

(Ω), we may ﬁnd a sequence of second-order tensors {G

} with

components in C

∞

(Ω) such that G

→ ∇u in L

(Ω). Then, from (XI.0.10 )

and Theorem XI.1.2, it follows that u satisﬁes the problem

∆u +

∂u

∂x

+ e

× x · ∇u − e

×u = u · A

+ ∇ep + F

∇· u = 0











a.e. in Ω

u = u

∗

:= v

∗

+ v

∞

at ∂Ω ,

(XI.4.2)

where A

:= ∇u − G

, F

:= v · G

+ f , and ep is deﬁned in (XI.1 .6). We

next observe that by the assumption on f and the fact that u ∈ L

(Ω), we

have F

∈ L

(Ω) ∩L

3/2

(Ω), for all k ∈ N. Set X

q,3/2

(Ω) = X

(Ω) ∩X

3/2

(Ω),

endowed with the norm k · k

q,3/2

:= k · k

+ k ·k

3/2

and let

M : (w, τ ) ∈ X

q,3/2

(Ω) → (z, φ) := M(w, τ ) ,

where (z, φ) satisﬁes

Possibly modiﬁed by the addition of a constant.

See ( X.6.6).

XI.4 Global Summability of Generalized Solutions when v

· ω 6= 0 771

∆z +

∂z

∂x

+ e

× x · ∇z − e

× z = w ·A

+ ∇φ + F

∇ · u = 0











in Ω ,

u = u

∗

at ∂Ω .

(XI.4.3)

In view of the H¨older inequality and the fact that u ∈ D

1,2

(Ω),

kw · A

≤ kwk

2s/(2−s)

< ∞, s ∈ (1, 2) , (XI. 4.4)

with the help of Theorem VII.8.1 we deduce, on the one hand, that the map

M is well-deﬁned and, on the other hand, that there exists a (unique) solution

(z, φ) ∈ X

q,3/2

(Ω) to (XI.4.3) satisfying the estimate (with q

= q, q

= 3/2)

k(z, φ)k

q,3/2

≤ c

i=1



kwk

/(2−q

)

+ kF

+ ku

∗

2−2/q

(∂Ω)



≤ c

k(w, τ)k

q,3/2

i=1



+ ku

∗

2−2/q

(∂Ω)



(XI.4.5)

Thus, choosing k such that

≤ 1/(2c) (XI.4.6)

and putting

δ := 2c

i=1



+ ku

∗

2−2/q

(∂Ω)



from (XI.4.5) it follows at once that M ma ps the closed ball { (u, φ) ∈

q,3/2

(Ω) : k(u, φ)k

q,3/2

≤ δ} into itself. Moreover, in view of the linearity o f

the map M and of (XI.4.6), from (XI.4.5) with kF

= ku

∗

2−2/q

(∂Ω)

0, i =

1, 2, we infer that M is a contraction, and therefore there exists one and only

one sol ution (z, φ) ∈ X

q,3/2

to (XI.4.3). Next, setting (w, τ) := (z −u, φ −ep),

we obtain

∆w +

∂w

∂x

+ e

×x · ∇w − e

× w = w · A

+ ∇τ

∇ · w = 0











in Ω

w = 0 at ∂Ω .

(XI.4.7)

Recalling that by the assumption on v and (XI.4.6), g := w · A

∈ L

3/2

(Ω),

with the help of Theorem VIII.8.1 we infer that the problem

∆

w +

∂

∂x

+ e

× x · ∇

w − e

w = g + ∇eτ

∇ ·

w = 0











in Ω ,

w = 0 at ∂Ω ,

(XI.4.8)

772 XI Three-Dimensional Flow in Exterior Domains. Rotational Case

has a (unique) solution (

w, eτ) in the class X

3/2

(Ω). It is easy to see that

(

w, eτ ) = (w, τ). Actually, the ﬁelds Z :=

w − w and χ := eτ − τ solve the

homog eneous problem (XI.4. 8) with g ≡ 0. Furthermore, Z ∈ L

(Ω), because

w, u ∈ X

3/2

(Ω), whi le u ∈ L

(Ω) by assumption. Therefore, from Theorem

VIII.8.1 we obtain Z ≡ ∇χ ≡ 0. Consequently, w ∈ X

3/2

(Ω), and so, again

by Theorem VIII.8.1 appl ied to (XI. 4.7), we obtain

k(w, φ)k

3/2

≤ c kw · A

3/2

≤ c kA

k(w, φ)k

3/2

Thus, using (XI.4.6) in this latter inequality, we deduce w ≡ ∇τ ≡ 0, that is,

(u, φ) = (v + v

∞

, ep + C), for some C ∈ R, and the proof of the theorem is

complete. ut

Remark XI.4.1 Note that Theorem XI.4.1 does not require the vanishing

of the ﬂux of v

∗

through the boundary ∂Ω. 

Remark XI.4.2 From Theorem XI.4.1 a nd (XI.0.10), it follows that the “ro-

tational term” e

×x ·∇(v + v

∞

) − e

×(v + v

∞

) belongs to L

(Ω). There-

fore, in parti cular, the results of Exercise VIII.7.1 apply to the component

(v + v

∞

) · e

= v

+ 1. 

Remark XI.4.3 If T = 0 (irrotational case), we k now that the summability

properties established in Theorem X.6 .4 in the signiﬁcant circumstance when

∗

≡ f ≡ 0 (XI.4.9)

(rigid body translating with constant velocity in a viscous liquid) are sharp.

More precisely, under the assumption (XI.4.9), we have (v + v

∞

) 6∈ L

(Ω) for

all s ∈ (1, 2] (see Exercise X.6.1), implying, in particular, tha t the total kinetic

energy of the liq uid is inﬁnite. It is probable that if (XI.4.9) holds, the same

conclusion can be drawn also when T 6= 0 (rigid b ody translating and rotating

with constant velocity in a viscous liquid), that is, the summabili ty properties

established i n Theorem XI.4.1 cannot be improved, and consequently, the to tal

kinetic energy of the liquid is still inﬁnite. However, no proof (or disproof) of

this statement is available to date. 

XI.5 The Energy Equation and Uniqueness for

Generalized Soluti ons when v

· ω 6= 0

An im mediate consequence of Theorem XI.4.1 is stated in the following the-

orem.

Theorem XI.5.1 Let Ω be a C

-smooth exterior three-dimensional domain,

and let v be a generalized solution to the Navier–Stokes problem (XI.0.10),

(XI.0.11) corresponding to the data

f ∈ L

4/3

(Ω) ∩ L

3/2

(Ω) , v

∗

∈ W

4/3,3/2

(∂Ω) , v

· ω 6= 0 . (XI.5.1)

XI.5 Energy Equation and Uniqueness when v

· ω 6= 0 773

Then v and the corresponding pressure ﬁeld associated to v by Lemma XI.1.1

satisfy the energy equation (XI.2.1).

Proof. Under the assumption (XI.5.1), from Theorem XI.4.1 we have, in par-

ticular,

v + v

∞

∈ L

(Ω) ,

so that the result follows at once from Theorem XI.2.1. ut

A signiﬁcant consequence to Theorem XI.5.1 is the foll owing result of

Liouville type.

Theorem XI.5.2 Let v be a generalized solution to the Navier–Stokes prob-

lem (XI.0.10), (XI.0.11) in R

corresponding to f ≡ 0 and v

· ω 6= 0. Then

v(x) = −v

∞

for all x ∈ R

Remark XI.5.1 It is not known whether the result of Theorem XI.5.3 con-

tinues to hold if v

· ω = 0. 

We shall next furnish suﬃcient conditions on the data for a corresponding

solution to be unique in the class of g eneralized solutions. The procedure will

be si milar to that adopted in Theorem X.7. 3 for the irrotationa l case. To this

end, we begin by proving the foll owing prelimina ry result.

Lemma XI.5.1 Let the assumption of Theorem XI.5. 1 be satisﬁed and sup-

pose, in additio n, that

f ∈ L

6/5

(Ω) ,

kf k

6/5

+ kv

∗

+ v

∞

7/6,6/5(∂Ω)

√

min

(

√

)

(XI.5.2)

with c = c(Ω, T , B) for R ∈ [0, B]. T hen, any generalized solution v to

(XI.0.10), (XI.0.11) corresponding to f , v

∗

, and v

∞

satisﬁes ( v + v

∞

) ∈

(Ω), along wi th the inequality

kv + v

∞

√

. (XI.5.3)

Proof. In view of Theorem XI.4.1 and the assumption on f, we have only to

prove (XI.5.3). To this end, let

M : (w, φ) ∈ X

6/5

(Ω) → (u, τ) ∈ X

6/5

(Ω) ,

where X

6/5

(Ω) is deﬁned in (X.6.5) and (u, τ) solves

774 XI Three-Dimensional Flow in Exterior Domains. Rotational Case

∆u + R

∂u

∂x

+ T e

×x · ∇u −e

× u = Rw · ∇w + ∇τ + f

∇ · u = 0











a.e. in Ω ,

u = v

∗

+ v

∞

at ∂Ω .

(XI.5.4)

It is convenient to endow the Banach space X

6/5

(Ω) with the “scaled” norm

k(w, φ)k

6/5

(Ω)

:= R

1/2

kwk

+ R

1/4

|w|

1,12/7

+ |w|

2,6/5

+ kφk

+ |φ|

1,6/5

Since by Theorem II.6.1, |w|

1,2

≤ c

|w|

2,6/5

, and by the H¨older inequality,

kw · ∇wk

6/5

≤ kwk

|w|

1,2

the assumption w ∈ X

6/5

(Ω) implies w · ∇w ∈ L

6/5

(Ω), and also

kw · ∇wk

6/5

≤ c

−1/2

k(w, φ)k

6/5

(Ω)

. (XI.5.5)

Consequently, from Theorem VIII.8.1, one deduces, on the one hand, that the

map M is well deﬁned, and on the other hand, that the following inequality

holds:

k(u, τ )k

6/5

(Ω)

≤ c



1/2

k(w, φ)k

6/5

(Ω)

+ kfk

6/5

+ kv

∗

+ v

∞

7/6,6/5(∂Ω)



(XI.5.6)

where c

= c

(Ω, B, T ). By the same token, for (w

, φ

) ∈ X

6/5

(Ω), i = 1, 2,

we ﬁnd that the corresponding (u

, τ

) satisfy

k(u

−u

, τ

− τ

6/5

(Ω)

≤ c

1/2

k(w

− w

, φ

− φ

6/5

(Ω)

i=1

k(w

, φ

6/5

(Ω)

(XI.5.7)

From (XI.5.6) it follows that M ma ps B

, the ball of X

6/5

(Ω) centered at the

origin and of radius δ = 1/(2c

√

R), into itself, provided

kfk

6/5

+ kv

∗

+ v

∞

7/6,6/5(∂Ω)

≤

√

and therefore a fortiori, i f

kfk

6/5

+ kv

∗

+ v

∞

7/6,6/5(∂Ω)

√

min

(

√

)

. (XI.5.8)

Furthermore, with the help of (XI.5.7), we conclude that M is a contraction

in B

. Thus, under the assumption (XI.5 .8), we may take w ≡ u in (XI.5.4),

and deduce that v

:= u −v

∞

is a generalized solution to (XI.0.10)–(XI.0.11)

XI.6 On the Asymptotic Structure of Generalized Solutions When v

·ω 6= 0 775

corresponding to the same data a s v. Also, from (XI.5.6) with u ≡ w, (XI.5.8)

and the fact that u ∈ B

, we obtain, in particular,

1/2

+ v

∞

≤ 2c



kfk

6/5

+ kv

∗

+ v

∞

7/6,6/5(∂Ω)



√

Since both v

+ v

∞

and v + v

∞

are in L

(Ω), as a consequence of Theorem

XI.4. 1 and the assumption on f , we inf er v = v

a.e. in Ω, and the proof of

the lemma is complete. ut

From Lemma XI.5.1 and Theorem XI.2.2 we immediately obtain the fol-

lowing uniqueness theorem for generalized soluti ons when v

· ω 6= 0.

Theorem XI.5.3 Let Ω be a three-dimensional exterior domain of class C

Further, let

f ∈ L

6/5

(Ω) ∩ L

4/3

(Ω), v

∗

∈ W

5/4,4/3

(∂Ω), v

· ω 6= 0.

Then if the data sati sfy (XI.5.2)

, v is the only generalized solution satisfying

these data.

XI.6 On the Asymptotic Structure of Gener alize d

Solutions When v

· ω 6= 0

This section is devoted to the study of the pointwise behavior of generalized

solutions to (XI.0.10)–(XI.0.11) when v

·ω 6= 0. Our main achievement is to

show that under the assumption of body force possessing mild regularity and

of bounded support,

every such solution v is pointwise bounded above by a

function that, roughly speaking, behaves like the Oseen fundamental solution

and consequently, exhibits a wake-li ke behavior. Analogous estimates are given

for ∇v and for the “m odiﬁed” pressure ep. However, it is worth emphasizing

that our analysis is not able to establish the existence of a leading term in t he

asymptotic behavior, which therefore is left as an open question.

We need some preparatory results, which begin with the following.

Lemma XI.6.1 Let v be a generali zed solution to (XI.0.10)–(XI.0.11) in

Ω := R

− B

ρ/2

, corresponding to f ≡ 0 and v

· ω 6= 0. Then, setting

u := v + v

∞

, we have, for all ε > 0,

|u|

1,2,B

≤ c R

−1+ε

, all R > ρ/2 ,

with c = (ε, R, T , v, p).

This latter assumption can be weakened to that f decays “suﬃciently fast” at

large distances. However, we will not state these more general conditions here.