2408 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

Take the derivative, multiply by xk (t,ω) , add, and integrate again in the usual way toobtain

12|un (t,ω)|2H −

12|u0n (ω)|2H

= −∫ t

0⟨A(un(s,ω)),un(s,ω)⟩ds−

∫ t

0

⟨N̂(un(s,ω)),un(s,ω)

⟩ds

+∫ t

0⟨f(s,ω),un(s,ω)⟩ds.

Recall that N̂ (u(t,ω)) = N (u(t,ω))+B(u(t,ω),q(s,ω))+B(q(t,ω),u(t,ω)) . From theabove lemma and that all functions are divergence free, we obtain∫ t

0⟨B(q(s,ω),un(s,ω)) ,un(s,ω)⟩ds = 0,

and ∣∣∣∣∫ t

0⟨B(un(s,ω),q(s,ω)) ,un(s,ω)⟩ds

∣∣∣∣≤C∫ t

0∥q(s,ω)∥V |un(s,ω)|2H ds.

Then one can obtain an inequality of the following form

12|un (t,ω)|2H +

∫ t

0∥un(s,ω)∥2

W ds

≤ 12|u0n (ω)|2H +C

∫ t

0∥q(s,ω)∥V |un(s,ω)|2H ds

+C∫ t

0∥f(s,ω)∥2

W ′ ds+12

∫ t

0∥un(s,ω)∥2

W ds.

Since t → ∥q(t,ω)∥V is continuous, it follows from Gronwall’s inequality that there is anestimate of the form

|un (t,ω)|2H +∫ t

0∥un(s,ω)∥2

W ds≤C (u0, f,q,T,ω) . (70.4.7)

The next task is to estimate ∥u′n(ω)∥L2([0,T ];V ′) for each fixed ω ∈ Ω. We will sup-press the dependence on ω of all functions whenever it is appropriate. With 70.4.6, thefundamental theorem of calculus implies that for each w ∈Vn,⟨

u′n (t) ,w⟩

V ′,V + ⟨A(un(t)),w⟩V ′,V +⟨N̂(un(t)),w

⟩= ⟨f(t),w⟩ .

In terms of inner products in V,(R−1u′n (t)+R−1A(un(t))+R−1N̂(un(t))−R−1f(t),w

)V = 0

for all w ∈ Vn. This is equivalent to saying that for Pn the orthogonal projection in V ontoVn, (

R−1u′n (t)+R−1A(un(t))+R−1N̂(un(t))−R−1f(t),Pnw)

V = 0