2418 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

It follows from equivalence of norms in finite dimensional spaces, the assumed esti-mates on M, A, and P and standard manipulations depending on compact embeddings thatthere exists an estimate suitable to apply Theorem 70.3.3 to obtain the existence of a solu-tion such that (t,ω)→ x(t,ω) is measurable into Rk which implies that (t,ω)→ vk (t,ω)is product measurable into V and H. This yields the measurable Galerkin approximation.

Also, the estimates and compact embedding results for Sobolev spaces imply an in-equality of the form

|vk (t)|2H +∫ T

0∥vk∥2

V ds+∥uk (t)∥2V ≤C (70.5.42)

where in fact C does not depend on ε,ω or k. Everything would work if C depended on ω

but because of our simplifying assumptions, we can get a single C as above.Next we need to estimate the time derivative in V ′. The integral equation implies that

for all w ∈Vk,⟨v′k (t) ,w

⟩V ′,V +

⟨Mvk +Auk +Puk + γ∗T F

((ukn−g(ω))+

)·

µ(∣∣vkT − U̇T

∣∣)ψ ′ε(vkT − U̇T

) ,w⟩

V ′,V

= ⟨f,w⟩ (70.5.43)

where the dependence on t and ω is suppressed in most terms. In terms of inner productsin V this reduces to(

R−1v′k (t) ,w)

V +

(R−1

(Mvk +Auk +Puk + γ∗T F

((un−g(ω))+

)·

µ(∣∣vkT − U̇T

∣∣)ψ ′ε(vkT − U̇T

) ),w)

V

=(R−1f,w

)V

In terms of Pk the orthogonal projection in V onto Vk, this takes the form(R−1v′k (t) ,Pkw

)V +(

R−1(

Mvk +Auk +Puk + γ∗T F((un−g(ω))+

)·

µ(∣∣vkT − U̇T

∣∣)ψ ′ε(vkT − U̇T

) ),Pkw

)V

=(R−1f,Pkw

)V

for all w ∈V . Now v′k (t) ∈Vk and so the first term can be simplified and we can write(R−1v′k (t) ,w

)V +(

R−1(

Mvk +Auk +Puk + γ∗T F((un−g(ω))+

)·

µ(∣∣vkT − U̇T

∣∣)ψ ′ε(vkT − U̇T

) ),Pkw

)V

=(R−1f,Pkw

)V

for all w ∈V . Then it follows that for all w ∈V,(R−1v′k (t) ,w

)V +