2422 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS
for each ε = 1/k, there exists a measurable solution to the integral equation
vk (t)−v0 +∫ t
0Mvkds+
∫ t
0Aukds+
∫ t
0Pukds+
∫ t
0γ∗T F((ukn−g(ω))+
)µ1/k
(∣∣vkT − U̇T∣∣)ξ kds =
∫ t
0fds (70.5.58)
where ∫ t
0
∫ΓC
ξ k ·wT dαds≤∫ t
0
∫ΓC
∣∣vkT − U̇T +wT∣∣− ∣∣vkT − U̇T
∣∣dαds (70.5.59)
Now for a given ω, the same estimate obtained earlier, 70.5.42 is available. Thus
|vk (t)|2H +∫ T
0∥vk∥2
V ds+∥uk (t)∥2V ≤C
where C is not dependent on k. Recall also that ξ k is bounded. Hence from 70.5.58, andthis estimate, it also follows that v′k is bounded in V ′. Thus
|vk (t)|2H +∫ T
0∥vk∥2
V ds+∥uk (t)∥2V +
∥∥v′k∥∥
V ′ ≤C
As earlier, we can take C independent of k and ω although we do not need this constant tobe independent of ω . Now for fixed ω, there exists a subsequence, still denoted as {vk}such that the convergences obtained earlier all hold, that is 70.5.46 - 70.5.53. Taking afurther subsequence, we may assume also that
ψ−h′1/k
(∣∣vkT − U̇T∣∣)⇀ 0 in L∞ ([0,T ] ,L∞ (ΓC)) ,
ξ k ⇀ ξ weak ∗ in L∞
([0,T ] ,L∞ (ΓC)
3).
That is, h′1/k
(∣∣v(1/k)T − U̇T∣∣) converges weak ∗ in L∞ ([0,T ] ,L∞ (ΓC)) to some ψ . This is
because
h′ε (r) =η2r√
r2η2 + ε
and this is bounded. Letting w ∈ L1([0,T ] ;L1 (ΓC)
),∫ T
0
∫ΓC
h′(1/k)
(∣∣vkT − U̇T∣∣)wdαdt
≤∫ T
0
∫ΓC
h(1/k)(∣∣vkT − U̇T
∣∣+w)−h(1/k)
(∣∣vkT − U̇T∣∣)dαdt
Thanks to the strong convergences and the uniform convergence of h(1/k) (r) to |ηr| ,∫ T
0
∫ΓC
ψwdαdt ≤∫ T
0
∫ΓC
∣∣η (∣∣vT − U̇T∣∣+w
)∣∣−η∣∣vT − U̇T
∣∣dαdt