70.5. A FRICTION CONTACT PROBLEM 2419
(PkR−1
(Mvk +Auk +Puk + γ∗T F
((un−g(ω))+
)·
µ(∣∣vkT − U̇T
∣∣)ψ ′ε(vkT − U̇T
) ),w)
V
=(PkR−1f,w
)V
Thus in V we have
R−1v′k (t)+PkR−1(
Mvk +Auk +Puk + γ∗T F((un−g(ω))+
)·
µ(∣∣vkT − U̇T
∣∣)ψ ′ε(vkT − U̇T
) )= PkR−1f
and R−1 preserves norms while Pk decreases them. Hence the estimate 70.5.42 implies that∥∥v′k∥∥
V ′ is also bounded independent of ε,ω and k. Then summarizing this yields
|vk (t,ω)|H +∥vk (·,ω)∥V +∥∥v′k (·,ω)
∥∥V ′ +∥uk (t,ω)∥V ≤C (ω) (70.5.44)
where C is some constant which does not depend on ε,ω, and k. Also, integrating 70.5.43,it follows that
i∗k
(vk (t)−v0k +
∫ t
0Mvkds+
∫ t
0Aukds+
∫ t
0Pukds+
∫ t
0γ∗T F((ukn−g(ω))+
)µ(∣∣vkT − U̇T
∣∣)ψ′ε
(vkT − U̇T
)ds)= i∗k
∫ t
0fds (70.5.45)
Where i∗k is the dual map to the inclusion map ik : Vk→V .Let
V ⊆W, V dense in W,
where the embedding is compact and the trace map onto the boundary of U is continuous.Using Theorem 70.5.2 and 70.5.1, it follows that for a fixed ω, there exist the followingconvergences valid for a suitable subsequence, still denoted as {vk} which may depend onω .
vk ⇀ v in V (70.5.46)
v′k ⇀ v′ in V ′ (70.5.47)
vk→ v strongly in C([0,T ] ,W ′
)(70.5.48)
vk→ v strongly in L2 ([0,T ] ;W ) (70.5.49)
vk (t)→ v(t) in W for a.e.t (70.5.50)
uk→ u strongly in C ([0,T ] ;W ) (70.5.51)
Auk ⇀ Au in V ′ (70.5.52)
Mvk ⇀ Mv in V ′ (70.5.53)
Now from these convergences and the density of ∪nVn, it follows on passing to a limit andusing dominated convergence theorem and the strong convergences above in the nonlinearterms, we obtain the following equation which holds in V ′.
v(t)−v0 +∫ t
0Mvds+
∫ t
0Auds+
∫ t
0Puds+