852 XII Two-Dimensional Flow in Exterior Domains
4cλ
−2(1−1/q)
(hu
1
i
λ,q
+ hui
λ,q
) < 1
so that (XII.5.61) implies w ≡ 0, thus completing the proof of the theorem.
ut
Remark XI I.5.4 It is worth observing that Theorem XII.5.1 does no t re-
quire that the datum v
∗
satisfies the zero-outflux condition:
Z
∂Ω
v
∗
· n = 0.
Remark XI I.5.5 We wish to emphasize that Theorem XII.5.1 furnishes ex-
istence (and uniqueness) for the physically significant problem obtained by
setting v
∗
≡ f ≡ 0, and describing the (plane) steady flow of a viscous liq-
uid around a cylinder translating with constant speed. Let us denote by P
this problem. For problem P, the assumption (XII.5.46) reduces, in fact, to
|log λ| > c, for some c = c(Ω) > 0, which i s certainly satisfied by taki ng λ
sufficiently small, which means, small translating speed. It is also interesting
to observe that the logarithmic factor, that i s crucial to prove such a result,
comes from the estimate of the total force, T = T (λ), exerted by the liquid o n
the cylinder in the Oseen approximation, and provided in Theorem VII.8.1.
As we know, in the class of generalized solutions to the Oseen approximation
T (λ) is not zero for all λ > 0, while it becomes zero in the Stokes approxi-
mation, that corresponds to λ = 0 (Stokes paradox, see Remark V.3.5). So,
we are expecting T (λ) → 0 as λ → 0, and Theorem VII.8.1 gives us an esti-
mate of the rate at which this happens. Consequently, if there were no Stokes
paradox, we would have not been able to prove existence for problem P!
Remark XI I.5.6 Solutions determined in Theorem XII.5.1 are unique in
the class of those sol utions verifying (XII.5.48). This result is much weaker
than the analogous one proved for the three-dimensional case. In fact, for the
situation at hand, we must require that both v a nd v
1
are smal l in sui table
norms, and it is not known if a solution v obeying (XII.5.4 6) is unique in
the class of those solutions v
1
that merely satisfy the condition hv
1
i
λ,q
< ∞;
cf. a lso Section XII.2.
Remark XI I.5.7 Si nce q < 6/5, solutions determined in Theorem XII.5.1
belong to D
1,s
(Ω), for some s < 2. Moreover, they are generalized solutions
in the sense o f Definiti on IX.1.1. In fa ct, they obviously satisfy conditio ns
(ii)-(v). Furthermore, since v ∈ D
2,q
(Ω) ∩ D
1,3q/(3−q)
(Ω), 1 < q < 6/5, by
Theorem II.6.1, it follows that v ∈ D
1,2q/(2−q)
(Ω) and so, noticing that
3q/(3 −q) < 2 < 2q/(2 − q)
we conclude, by interpolation, that v ∈ D
1,2
(Ω) and so also issue (i) of Defi-
nition X.1.1 is verified.